iserith
Magic Wordsmith
We're a bit hit and miss for inspiration. I know that when I DM, I really, really need to up my game and award it more often. It just tends to get forgotten.
The main issue with inspiration though, like a lot of fuzzy, DM arbitrated rules, is that the DM has so many plates in the air, it's easy to forget. The DM isn't deliberately short changing anyone, it's just that it can be hard to remember, "oh, hey, yeah, this character has that trait, so, he or she should get inspiration for doing that thing". I mean, good grief, I have enough trouble remembering the names of the characters my players play, so, trying to remember that Bob has Trait X is often very far down on the list.
Which is, honestly, why I don't see inspiration as a good replacement for feats for rewarding play. With feats, it puts the onus on the player to deal with it. You want your character to be especially tough? Good, take a feat that bumps your HP. Don't expect me, as DM, with four other people around the table plus my own NPC's and whatnot to deal with, to remember that three months ago you told me that your character was a "tough guy" and you should be getting inspiration for playing that up.
All the DM need do is tell the players how they can get it in the game, which can reasonably include claiming it for themselves. Believe me, players have a vested interest in remembering. They will make sure they play to their character's personal characteristics to get that Inspiration. If the DM can't bring himself or herself to remember, outsource it.
I don't use feats in most of my games. I don't use them in my one-shots and didn't use them in two of my three last campaigns. In my current campaign, their availability is gated behind faction membership. But I do use Inspiration in all of them as described. And it works for incentivizing particular behaviors.