You know what's gritty? It's not sitting around waiting for your HP to return, or relying on magic to keep adventuring. It's going adventuring in spite of the risks, and there's nothing that makes a player not want to take risks than their PC being low on HP.
So... think of it this way: HIT POINTS ARE GRIT. When you wake up in the morning with full HP, it's not saying that you're not injured. It's saying that you're gonna go adventuring today IN SPITE of any pesky injuries. Whatever "healing" you received - be it bandages, magic, or whatever, was enough to make it so that you feel well enough to adventure. When you take damage later, it may just be that you opened yesterday's wounds, rather than getting new ones. Now that's GRIT.
If that's not good enough for you, I recommend doing that anyway, but add a wound condition track, or a chart, or just use the exhaustion mechanic. That way you can have full HP (enough to make a cautious player still, y'know, play) but with a penalty that they can live with. (The OneD&D exhausted condition is simple for this: You have "Exausted X" (where x is accumulative, and represents a -1 to all "d20 Tests" (aka d20 rolls). It's not as bad as disadvantage, so it's less likely to result in player-timidity.)
Heck, I might just write this up in a fuller post and flesh it out.