Let me start with a pragmatic answer that tries to establish exactly how wisdom overcomes ignorance and provides a true antidote to suffering.
Consider the famous allegory of the Snake and the Rope:
Two friends walk along a grassy meadow on a moonlit night. Suddenly,
one of the friends gives a jump and yells, “snake!” The other looks in
the same direction and sees only a coiled rope. She begs her friend to
look closer. He does and now likewise sees only a coiled rope and
gives a little laugh of relief as his heart slows.
Think about what is happening here. When the friend that mistakes the rope for a snake looks upon the meadow and knows "snake" his heartbeat instantly begins to rise. His fear appears to arise near instantaneously with the knowledge that there is a snake before him. His fear doesn't arise because he first stops and conceptualizes that "I" am a permanent being or a unitary being or an independent being and this snake here is threatening "me". Similarly, his fear does not dissipate upon conceptualizing that "I" am not a permanent being or a unitary being or an independent being.
What actually happens is that his friend begs him to look closer and he discovers that no snake exists at all and seemingly near instantaneously his heart begins to slow. Upon understanding that what he is seeing is a mere illusion of snake his fear is completely dissipated. This is how wisdom overcomes ignorance. Ignorance stands no chance whatsoever against this incontrovertible knowledge that no snake whatsoever exists in that meadow.
The point is that the wisdom knowing that the self is not a permanent, unitary, or independent being while having some efficacy is insufficient to overcome all our afflictions. It is a tourniquet in a situation calling for a real cure. Still, it is useful to apply tourniquets! Applying a tourniquet can save the patient waiting for a real cure.
Here is another well known famous allegory which makes the same point from Insight into Emptiness page 256:
For example, a young woman may want to have a child. When she is
asleep, she dreams she gives birth to a child and is elated. But later
in the dream, the child dies and she is devastated. However, on
waking, she sees that neither the exhilarating appearance of having a
child that brought her joy nor the horrible appearance of the child’s
death that caused her anguish is real.
Does the young woman's anguish go away when she wakes up upon contemplating that she is not a permanent, unitary and independent self? Did she contemplate that the "mine" making was inappropriate with regard to the child because the self of her person was subject to causes and conditions? No, that is not what happened. What actually happened was the wisdom knowing that it was all just a dream overcame her anguish and elation. The anguish and elation she felt could not withstand the certain knowledge that it was all just a mere dream. This is how wisdom overcomes ignorance leading to the cessation of suffering.
I've noticed this in my own life. For a long time I pondered what the difference was between "reality" and a "dream." I was certain that there was a difference. It seemed so obvious and something deep inside of me revolted against the idea that all of this is like a mere dream. And then one day I recognized that this very revulsion or inner objection was innate ignorance that has been with me since beginningless time. You can see it replete on this very forum. So many argue and revolt against the idea that phenomena are unreal or like a dream. That the world lacks essence. That all exists as mere convention. They argue that the question is entirely irrelevant and yet strenuously object at the same time that phenomena are unreal seeing no tension in these two things. Paraphrasing:
That question is irrelevant and holds no practical consequence, and oh btw, obviously the world is real and the world is not an illusion and to say otherwise is silly nonsense!
If it was truly irrelevant or immaterial, then no one would object or find it deeply unsettling this idea that the world might actually be just like an illusion holding no objective reality whatsoever. No, the objections come from innate ignorance which can't imagine such a thing. When innate ignorance tries to imagine it we usually fall into nihilism thinking that if the world is illusion, then nothing at all exists or matters and hence destroy ourselves mishandling the snake of emptiness.
Now, for authoritative texts backing this up. First, have a look at Je Tsongkhapa in his Great Treatise Volume 3 page 196 in the Chapter Not Negating Enough:
Opponent: The object to be negated is an intrinsic nature that has three attributes: (1) causes and conditions do not bring it into
being, (2) its condition is immutable, and (3) it is posited without
depending on some other phenomenon.
...
Fallacies arise if we follow this opponent’s interpretation. Since the
partisans of non-Madhyamaka Buddhist schools have already established
that compounded phenomena are produced by causes and conditions and
are mutable, we should not have to demonstrate to them the absence of
intrinsic nature. They also should have recognized that things lack
intrinsic nature. So how can this be the unique Madhyamaka object of
negation?
Many Madhyamaka texts adduce arguments such as: If things existed
essentially, then they could not depend on causes and conditions, they
would have to be immutable, and so forth. However, these statements
indicate fallacies that would be entailed if things existed
essentially; they do not identify the object of negation on its own
terms.
It is the case that if something existed ultimately, existed in reality, or truly existed, then it could not depend on causes and
conditions, and so forth; however, that is not what ultimate existence
means. For example, even though being a pot entails being impermanent,
impermanence is not the proper meaning of pot; rather you have to say
that it means a “bulbous splay-based thing able to perform the
function of holding water.”
Likewise, if something existed ultimately, etc., it would have to be a
partless thing; still, here in Madhyamaka we do not suggest that
“partless thing” is the fundamental object of negation. Since partless
things are merely imputed from the unique perspective of advocates of
philosophical tenets, such notions are not the funda- mental cause
that binds embodied beings in cyclic existence. Fur- ther, even if you
determined that those partless things lack intrin- sic nature and then
meditated on that, this would not at all counter the ignorant
conception which has operated from beginningless time. Therefore, even
optimal and direct knowledge of that would not overcome the innate
afflictions.
On page 197 it continues after a bit:
It would be extremely absurd to claim that you can overcome innate
afflictions by seeing as nonexistent the twoselves imputed by acquired
misconceptions. Candrakırti’s Commentary on the “Middle Way” says:
"When knowing selflessness, some eliminate a permanent self, But we do
not consider this the basis of the conception of “I.” It is therefore
astonishing to claim that knowing this selflessness Expunges and
uproots the view of self."
Also, Candrakırti’s Explanation of the “Middle Way” Commentary says:
To elucidate this very point, the irrelevance of such to innate
afflictions, by way of an example:
Someone sees a snake living in the wall of his house.
To ease his concern, someone else says, “There is no elephant here.”
Alas, to others it is ridiculous
To suppose that this would dispel the fear of the snake.
Candrakırti refers to the selflessness of the person, but it is the
same for the selflessness of objects; he could have added:
When knowing selflessness, some eliminate an acquired
conception of self,
But we do not consider this the basis of ignorance.
It is therefore astonishing to claim that knowing this selflessness
Expunges and uproots ignorance.
So where does that leave (some) Theravada adherents view of anatta that holds that the self of persons to be a permanent, unitary and independent is the only thing to be refuted? To be clear, the view of the person as a permanent, unitary and independent being is an acquired ignorance and not innate. This view to be refuted, in the time of Shakyamuni Buddha was propounded by other religions as atman. In modern non-Indian religions it is analogous to the soul. This is something that is a learned ignorance and not innate. It has not necessarily been with us since beginningless time.
In modern Western society many, many, many, many non-Buddhists have given up this acquired ignorance and do not believe in the soul. Lots of western scientific practictioners are avowed atheists such as Daniel Dennett. Dennett does not believe in a permanent, unitary and independent self anymore than most Buddhists, but I do not think he is enlightened nor has he overcome the afflictions. I don't think Dennett has achieved the soteriological goal of Buddhism. Why? Because the wisdom overcoming this acquired ignorance does not stand a chance against the innate ignorance thinking the world and things objectively exist in and of themselves. Thinking that the world and things are not like illusions and exist as more than mere convention. It stands no chance against that ignorance. BTW, merely having a correct inferential understanding that the world is like an illusion and all things exist as mere convention is also not enough to overcome this innate ignorance. No, to overcome it we must directly see the lack of inherent existence.
In summary, with respect, I think it is entirely useful and correct that Theravada adherents do not believe in a permanent, unitary and independent self. It is necessary to overcome this acquired ignorance in order to progress on the path to overcoming innate ignorance. However, stopping with overcoming this acquired ignorance is insufficient to pull up our innate root ignorance that has been with us since beginningless time or to truly stop the afflictions.
I believe the Pali Suttas contain the core teaching that goes beyond overcoming this acquired ignorance and shows the way to overcoming innate ignorance, but that the Pali Suttas do so in brief. Thus, someone following the Theravada which only relies upon the Pali Suttas, but understands them correctly and sees how they can overcome this innate ignorance can become Arahants which is the emphasized path in Theravada.
Hope this helps!
Addendum 1:
Some people naively think that Buddhism is all about not going to extremes. That moderation is a virtue in and of itself. Indeed, the Buddha himself espoused the Middle Way free from the extremes of eternalism (Everything exists!) and nihilism (Everything doesn't exist!), but I think it is important to understand that the Middle Way is also an extreme and that the Buddha often espoused us to go to the extreme. What can be more extreme in this world than the perfectly perfected state of the Supreme Buddha? The Buddha exhorted us time and again to push ourselves into perfection. The Buddhist path itself could be thought of as the quintessential guide into going into an extreme perfection. What do I mean when I say the Middle is also an extreme? Well, I think ever deepening the wisdom realizing emptiness pushes us into the knife edge balance between eternalism and nihilism. It is a high wire act that has precarious drops on both sides. If we don't negate enough we risk falling into eternalism. If we negate too much we risk falling into nihilism. Of the two, the latter is said to be a greater risk. But make no mistake Buddhism is all about going to the extreme: perfecting ourselves in order to help others via the Middle Way.
