Yikes.
Players are free to ignore any of the following and only engage when it benefits them:
- languages they speak
- armor proficiencies
- weapon proficiencies
- tool proficiencies
- spells they can cast
- rituals they can cast
- features they can use after a rest
- skill proficiencies
- equipment they carry
- wealth they have
- services and hirelings
- mounts they can ride
- activities they might undertake while traveling
- downtime activities
- available combat actions
I'd hate to point at any of these in a given situation where they might be useful for fear of reminding anyone how to play their character (gasp!).
No, they are not, at least not for all of those. If you ignore armor proficiencies, there is a strong feedback for doing so -- you are hit more often or you have disadvantage on lots of things. If you ignore a tool proficiency, when you take an action that can use that tool, you are hit with a strong feedback -- you do not get to add proficiency bonuses to your roll.
Most of these things have feedbacks, in fact, whereas ignoring your BIFTs has no negative feedback at all. This is a terribly muddled list that doesn't consider how these things are actually incentivized directly or penalized directly in the game. Sure, you can ignore them, but then there is a feedback for doing to, almost always negative.
I'm trying to imagine the tables you're sitting at because all of the conflict you describe is just alien to my experience with the game. Who is it that behaves so terribly? I'd honestly like to know so I can avoid them should I encounter them in the wild.
I'm playing in a Blades in the Dark game right now, which features strong characterizations effects. So far, my rival has caused massive problems for my crew (the other players), and I've said bupkiss about it despite knowing exactly what's going on. This is because it's related to my backstory, which I do not share, because it involve my character having betrayed a former team for personal gain -- something I might do again but even if I don't, the implication is there.
Another example, my character has the reckless example. My character has done some incredibly stupid, very not good for anyone, things in the name of being reckless. This has blown up at least one, and seriously complicated many, things we've done. Hell, at one point, the crew had a threat pretty much locked down and I screwed it up entirely with a reckless move (had it paid off, it would have been glorious, but the odds were extrememly against that). This cause harm (injury) to two crewmates, almost ruined the score, and has had lasting effects as we didn't deal with the threat and had to spend lots of our downtime resources countering it - plus it showed up twice in other scores to cause further complication.
But, since Blades isn't a game where the team concept is placed first, but rather the characters, and it fully allows very different agendas from the characters (but not as much as Apocalypse World, due to the Crew mechanics), this isn't a problem at the table, even as it's a major problem for the characters and crew.
However, you're right, this would be extremely odd in a D&D game, because the first thing is the party and you're expected to bend character to this.
Respect is the thing your character believes in most strongly, though they're extremely hot-headed and someone else can use that to exploit them or cause them to act against their best interests.
Where does it say your ideal is, in any way, a stronger force for your character than their flaw, or bond? And, no, there's no way to exploit or cause them to act against their best interests -- at all.
There's no angel versus devil dichotomy. You've described someone who will smack you up if they see you disrespecting a gentle soul, but will only use the amount of force necessary to straighten you out.
I have an ideal and a flaw that are at odds, this is absolutely an angel on one shoulder and a devil on the other. I might have a Bond of "family first" and a flaw of "drunkard" and spend far more time drunk, not being there for my family, but when I'm sober I am. I mean, that's like super tropey right there. There's nothing that requires BIFTs to be in harmony. Even your example is one where the character is choosing to enact violence when they could show respect and discuss and educate rather than bash. And, how much force is necessary to straighten someone out? What if the first beating doesn't take?
Inspiration is everywhere when campaigns first start because it helps to get everyone on the same page about the game we all want to be playing. As time goes on, it really only makes a showing in those awesome moments where something cool happens and there's magic in the air.
My big takeaway from this thread is that I'm incredibly lucky to play with the people I play with. There's so many magical moments!

Yes, no one else's game has magical moments.
I'm simply asking you to reconsider in light of D&D not being any of those other games. It's clear that you have a fixed mindset on this, so I won't try and push your thinking anymore.
There are quite a lot of people that have never even heard of those games that find BIFTs useless or harmful. There's threads here about it, and it's a common comment. The awareness of other games that actually showcase how this stuff can work very well just goes towards understanding exactly how BIFTs fail to do what they do. Frankly, given the nature of 5e as a GM-centered game, the attempt was ill-conceived to begin with -- it just adds to GM overhead and is one more thing that you have to hope the GM does for you.
I've not encountered this argument about 5e stripping power from the DM, but we'll have to discuss that in a different thread.
Edit: Formatting
Here I'm with you -- I'm not sure at all what
@tetrasodium is saying, here. TS -- maybe start a thread on this idea?