I played around with a similar idea when I was trying to figure out how to make 'players roll all the dice' more literal and not have them roll for enemy actions - enemies would do fixed damage, and you would roll to see how many hp you got as needed.

It worked like this: your base hp was one maxed out hit die. As you took damage, you would track damage separately from hit points. If you had more damage than current hp, you would roll Hit Dice (with con, no action). This was added to your hp. Once you had more hp than damage, you stopped rolling. If you ran out of HD and still had more damage than hp, you were dying.

Healing removed damage, and you could get back HD on a short rest (but no more than your current level per day, creating a second new number to track.

I never got around to playtesting this.

If I wanted to incorporate the 'key ability instead of con' idea to the mix, I might reduce the number of HD you can get back.