Inspiration does not, at all, do anything for narrative play, yes.
Rewarding playing of BIFTs is about as useful to narrative play as using your face as a wagon wheel, which is to say some very small amount and not recommended.
I thought so, too, on first read, but then my experience with the system -- trying to use it to do these things, and even making changes to enhance it's capabilities -- showed me that it doesn't work at all. It has no teeth, what it does is cosmetic at best, and it doesn't encourage any real changes in the fiction. It doesn't do this because the GM has no way to invoke BIFTs with the rules, and there's no requirements on the player as to how a BIFT actually functions. Instead, the only thing in the rules is that the GM may grant Inspiration to a player that roleplays a BIFT well. Okay, that's terrible, it does nothing to encourage any play because the trigger is in the GM's grey matter, so I, as a player, can feel that I just knocked it out of the park and the GM might not even notice, much less agree. That's not a virtuous reward cycle.
No, what you're talking about is using this terrible subsystem in a way the system doesn't advocate, which is the GM engaging tightly with the BIFTs and encouraging playing to them and pre-emptively offering Inspiration. Except, this isn't the system in the book -- it's the GM doing their own thing. You can tell this because it if were more aggressive, you'd be hearing all about it in thread from people that dislike that kind of thing (seemingly a large and vocal plurality). "I have to offer my players metacurrency to just play their characters!!?!" Yeah, it's not there.