On the topic of fixing the 'default'/built in method of inspiration (Not to say that putting in more ways to get it is a bad way to approach it!), specifically points 1 & 2 of your 'problems' section:
Would it allow you to avoid having to remember as much/evaluate in as much detail if you simplified the existing rules even further? It seems like inspiration could be boiled down to rewarding players for having a strong character moment- either an establishing one or one that makes you go "Yeah, that scene/action/moment was totally [insert character name here]".
You could theoretically, as the DM, ignore the written bonds/flaws/ideals yourself and leave them as a guideline/reminder for the player and look out for just those strong character moments (which hopefully would be notable on their own without having to do any active analysis/reminding yourself of "Oh, that was in line with his written traits").
(On a side tangent: In my personal experience/opinion, I don't entirely like the idea of using the prebuilt ideals/bonds/flaws. It's fine as a set of guidelines for doing your own custom ones, but I worry that the boilerplate ones end up feeling unnatural/shoehorned for the PCs and make people either explicitly call them out or forget/ignore them. I think writing custom ones based on what you actually have in mind for your character makes it easier for them to feel right and be used naturally, even if it means leaving them blank until you figure out who your character is if you're the type of player who doesn't determine that in advance)
It just seems like it'd be relatively simple to make those "That was awesome!" moments that you guys see all the time into "That was awesome! Have inspiration." while keeping the spirit of the original Inspiration guidelines, even if you have to put an artificial cap on it per person/session or something.
Admittedly, I may be wrong on that and it's still just not simple to think "That's worthy of inspiration!" in the moment, or they may not be as clear cut as I think they are.