Then give your player a Death Heart and make him spend HP to move, act, and roll dice. You can put that HP damage on an Effort scale too or offer an interesting dilemma.
A neat idea - basically making HP your action economy (I know there’s a game that does this, but I can’t remember what it is right now - it’s a space bounty hunter game), but that doesn’t really reduce the number of point pools players and GM are being asked to track, so different mechanic, same problem (IMO). Still, try it out! Tell us what happens!
How about this for an idea? Makes dropping to 0 more dramatic and finding somewhere safe to rest a priority after someone has been knocking on death’s door.