Does the question "what are the pitfalls to improvement?" further the purpose (to improve)?
It depends. In the simplest case, yes of course - and Buddha talks a lot about this in the Pali Canon: For any obstacle Xi: what is the exact nature of Xi? what are the necessary and sufficient factors required for Xi to arise? which factor Xi-1 of these factors can we remove to prevent Xi from arising? (Repeat recursively as needed for the factor Xi-1 in place of Xi, and then its necessary factor Xi-2 etc. until X0) Finally, what is our exact plan of attack to uproot the whole chain?
Now, as Mahayana practitioners figured out, the desire to advance on the path can itself be a major pitfall. This is even recognized in the Pali Canon itself, which says that udhacca-kukkucca (restlessness and anxiety about one's Dharma practice) is one of the fetters that must be eradicated on the advanced phases (along with arrogance & conceit on the other extremum).
My favorite illustration of this is Dutiya Anuruddha Sutta (AN 3.128 or 3.130 or 131 depending how you count):
Then venerable Anuruddha came to venerable Shariputra and said:
-- As much as I, friend Shariputra, look at things with purified Heavenly Eye, see things with thousandfold-superhuman vision, train in confidence unmovable, in awareness undimmable, in tranquillity imperturbable, and in focused mind unscattered, still I can't "having stopped feeding the asavas, achieve liberation of mind".
-- All these dharmas, friend Anuruddha, of the kind "I look at things with purified Heavenly Eye, I see things with thousandfold-superhuman vision" -- this is your conceit.
And the dharmas like "I train in confidence unmovable, in awareness undimmable, in tranquillity imperturbable, and in focused mind unscattered" -- this is your udhacca.
And the dharmas like "still I can't "having stopped feeding the asavas, achieve liberation of mind" -- this is your kukkucca.
You'd better, friend Anuruddha, these three dharmas abandon; To these three dharmas not attending, on the deathless aspect focus your mind.
In this scenario, worrying about one's improvement was actually itself an instance of "feeding the asavas".
Compare this with the following verses from Prajnaparamita-Ratnagunasamcayagatha which says:
Also, not the position of body, nor of feelings, nor thoughts, nor volitions,
Nor of consciousness -- position even a little does not bother him.
Thus, not getting stuck on any single dharma, without a "home",
Not clinging, he attains Enlightenment of the Lucky Ones.
Having stopped again and again wavering searching for wisdom,
"Whence?", "Why?" -- of such dharmas empty,
Having stopped getting upset, caught up, or scared,
Soon he can reach Enlightenment, such Bodhisattva.
So yes, inquiring into the pitfalls to improvement does further, including inquiring into these notions of "inquiring" and "improvement" themselves.
P.S. and what is that "deathless aspect" (amrita-dhatu) that Shariputra was talking about? As is said in KN Ud 1:10,
Bāhiya, you should train yourself thus: In reference to the seen, there will be only the seen. In reference to the heard, only the heard. In reference to the sensed, only the sensed. In reference to the cognized, only the cognized. That is how you should train yourself. When for you there will be only the seen in reference to the seen, only the heard in reference to the heard, only the sensed in reference to the sensed, only the cognized in reference to the cognized, then, Bāhiya, there is no you in connection with that. When there is no you in connection with that, there is no you there. When there is no you there, you are neither here nor yonder nor between the two. This, just this, is the end of stress.