Suppose an ordinary being thought the following:
“My friend purchased a new car the other day. That car is useful for
them to get from point A to point B. That car is theirs. That car
would also be useful for me to get from point A to point B. However, I
should not take that car as it is theirs and not mine."
Now, after this ordinary person wins stream entry would they automatically abandon the above thought or perhaps not need it anymore? Isn't this question similar to the OP? I think the answer is clear: the person winning stream entry would still have these discursive thoughts and if they abandoned the conclusion - thinking that it is now permissible or skillful to take the car - they'd be doubly wrong. If they thought that because 'I' and 'mine' were self-views that they no longer believe in or that they have outgrown these childish notions and that it was no longer skillful to refrain from taking the car, they'd be doubly wrong. If someone thought they'd won stream entry while holding the above notions they'd be dangerously wrong.
"Therefore, would a stream enterer also automatically abandon the following skillful reflection or perhaps not need it anymore?"
An Arya being is no longer fooled that the self is truly existing, however an Arya being does not come to the false conclusion that this knowledge refutes the conventional existence of the self. Therefore, emphatically no, an Arya being does not give up skillful means that are based on conventional truths. An Arya being is not held sway to the illusion that this reflection (a conventional truth) has true or ultimate existence. That is the point.
The Buddha did not argue with the world. The Buddha did not reject conventional truth. Upon perceiving the ultimate truth he did not use this to refute the conventional. Doing so would be an error of falling to the extreme of annihilationism.
When you analyze dhammas with reasoning trying to find a self and come up with nothing... to come away from this thinking that the dhammas do not have conventional existence is an error. It would be like analyzing a chariot and trying to find the true existence of the chariot ... is it in the parts? is it in the whole? is it both? neither? and when you come up with nothing ... would you conclude that chariots just don't exist at all? That they have no use in this world since they don't truly exist and you've "outgrown" them??
This question is like asking if an Arya being, after discovering that chariots do not ultimately exist, would use this to deny the conventional existence of chariots and therefore refuse to ever get in one again - preferring to walk instead!
It might be good for people having doubts about this to meditate on this question and read Nagarjuna's 24th chapter of the MMK where he specifically discusses what the doctrine of emptiness implies for the Four Noble Truths - and maybe more importantly - what it emphatically doesn't.
The real danger with this whole line of reasoning - mistaking the result of analyzing for the ultimate as refuting conventional truth - is that nothing whatsoever can withstand ultimate analysis. That is, if you ask whether anything at all has true existence - a pot, a headache, a chariot, a car, persons, evolution, special relativity, quantum mechanics, basic arithmetic, electricity - nothing at all can withstand this analysis. You'll come up empty each and every time. So if you then conclude that pots, headaches, chariots, cars, persons, evolution, special relativity, quantum mechanics, basic arithmetic, electricity are all conventionally non-existent or conventionally false ... boy will you have a hard time living in this world and communicating with your fellow sentient beings. It is just an error. I'd say go back and look again :)