In theory, there is mechanical carrot for playing BIFTs called inspiration. In practice, it rarely happens at most tables.Sorry, but this isn't actually having that character DO anything, it's just play-acting. Which is one of my complaints about these systems -- they don't encourage actual changes in play, they encourage superficial changes at the play-acting level. Because, if we analyze that scenario for actual impacts, what happens is the character meekly goes along with the group, but makes some funny noises along the way. This isn't actually engaging a flaw, it's just taking a prompt to do some play-acting and pretend we've done something impactful with our character.
And, to be perfectly clear, this is fine. Honestly, it's the expectation at a normal D&D table and it's entertaining, so there's nothing bad here. The point I'm making isn't that this is weak-sauce or terrible, because it's not, but that it's not actually engaging in playing a character that has a flaw in any meaningful way. Again, you don't have to, but the claim is that BIFTs do this -- they engage character in meaningful ways. This example shows this to be strongly overstated -- there's nothing meaningful here outside the entertainment variety. Now, D&D is structurally a team game where you have to belong to the group consensus to advance play (broadly speaking), so having systems that actively promote characterization is something that is a hinderance to this. BIFTs, as the tacked on kluge they are, try to suggest that you do get some individuality in D&D, but they don't provide any way to actually do this, so the default is that you're expected to use BIFTs in service of the team goal approach, not against it. This rips the meaning out of them, and they become nothing but glorified prompts from occasional play-acting and not actual characterizations.
However, personally I never remember them because there's too much going on with 6 PCs at my table. I guess I could have players claim them, but I'm not sure I want people thinking of what their PC does in terms of what will get them inspiration.