Like many 5e players, I investigated other systems during the drama in January. I have converted my table over to Pathfinder 2e, so that's what I'll mostly be running going forward.
However, I'm not one of those people who thinks 2e is better than 5e in all respects, and I'll probably run 5e again at some point in the future. And when I do, there's a mechanic from Pathfinder that I will 100% be porting over to 5e: Hero Points.
Below I will provide a version of this mechanic, written in the mechanical language of 5e (to the best of my ability)
Hero Points
(replaces inspiration)
Hero points represent a player character's unique ability to act heroically against adversity.
At the beginning of each play session, each player loses any hero points they have accumulated, then recieves 1 hero point. Depending on the average duration of your group's gaming sessions, the DM may increase or decrease the frequency of this by periodically resetting hero points mid-session, or only once every couple sessions. The expectation is that this reset will occur roughly once every 3-5 hours of play.
The DM may award additional hero points for any reason. They are encouraged to give out hero points to reward good roleplaying, good tactics, or any behaviour which improves the play experience for everyone at the table. A DM should award an average of approximately one hero point for every hour of play. You can have multiple hero points, but you can never have more than 3 hero points.
Whenever you roll an attack roll, saving throw, or ability check, you may spend a hero point after seeing the result and after the DM tells you whether or not the roll was a success or failure. If you do, you reroll the check and must use the new result, even if it is worse. This roll is unaffected by whether you have advantage or disadvantage on the check.
Whenever you would fall unconscious due to being reduced to 0 HP, or whenever you would accumulate a death saving throw failure, you may spend all of your hero points in order to remove any death saving throw failures and become stabilized at 0 HP.
The DM may provide other options for how to spend hero points, such as altering the story or introducing new information which benefits the party.
That's it, that's the whole mechanic. In general I just think 2e does a much better job of the "inspiration" system than 5e.
In my years as both a player and a DM, I can count on one hand the number of times I ever saw Inspiration being awarded to 5e players. It simply isn't a good system. The DM fiat needed to receive it makes it unpredictable, and it's way too hard to get players to actually spend it once they have it.
The Hero Points system is much better because it resets every session, meaning that players will have a predictable way to get hero points, and that they will have a good reason to spend them; even if the DM forgets to award it. And since they will be spending that resource more often, the DM will have more reminders that the mechanic exists.
I encourage DMs out there to experiment with this system in their own games. And let me know how it goes.
Aside from the death save thing, that's really just a minor tweak on the basic mechanic of 5e's inspiration (rerolling after the fact instead of getting advantage beforehand), plus letting them stack when accumulated. The biggest problem with 5e inspiration being hoarded isn't that it doesn't stack, it's that many DMs just don't give it out with any frequency whatsoever even though the system's description encourages DMs to give it out for all the reasons mentioned (cool roleplay, clever tactics, etc). The idea is supposed to be to encourage players to "go ham" getting into character and immersing themselves into the game to earn it back after using it, not to lack such motivation because they feel safe with three do-overs in their back pocket.
The system isn't flawed, it's just underused and the slightly different version of it doesn't change anything unless the DM/GM actually gives it out. And I would think players would be more likely to ause inspiration or hero points or drama dice or epicness metanuggets or whatever you want to call it if they heard the DM say "I would give you inspiration for that but you haven't used the one you already have" than they would if they didn't "waste" that extra one by just tossing it on the pile they already have "in case of emergency" like an offline FPS player hoarding their rocket ammo but never using their most powerful weapon because they might need it all against the next tough enemy.
And I'm not saying either slight variation of the same basic mechanic is necessarily better. I'm saying that the solution the problem of it not getting used will remain the same in both unless the DM/GM takes the exact same actions with either (actually handing it out). Again, the problem isn't the system, it's operator error.
I feel like you maybe missed the most important part, which is that each player gets reset to 1 hero point at the beginning of each session. If you have 2 or 3, you lose the extras, whereas if you had 0 you get one for free. So they have incentive to spend them, since they literally can't hoard them. And as the game is making sure they get at least one every session and giving them incentive to have as few remaining as possible at the end of the session, they will always be thinking about spending them. And if they are spending them often, the DM is more likely to hand out additional points, since they will have more regular reminders of their existence.
You can just tack on that one thing about getting it for free at the beginning of a game without calling it something different. It's still functionally the same system, not some "new" mechanic. And if the person running the game doesn't keep pointing it out the players are not going to pay any more attention to it than before. Tacking on a slight addition like that will not "fix" a system if people still don't actually use it.
The problem is operator error, not design. An automatic transmission won't make a car easier to drive for a person who keeps forgetting that you have to actually press down on the long foot pedal on the right side to make it go. Much like the majority of PF, this "different mechanic" is just D&D dressed up in a new T-shirt with a different scent of perfume.