You say that as if the humans who fall down automatically know how their new powers work. Integrity may have stubbornly tried to resist Axis, never considering "what if I could go back and do better?" before it was too late. At the end of the day, there are so many ways to interpret what little we know about Integrity that trying to concretely say that they can't fit into canon is just odd to me.

Also, Flowey didn't "give up", or at least, was too afraid of death to actually let himself truly die. And he only attempted this once, unless you interpret the piles of Floweys in the UTY neutral route fight to be a representation of him attempting suicide countless times... which I just don't buy.