Here is what I'm probably going to have to do:
1) You can only expend HD during a short rest by consuming a healer's kit. IIRC, this was the case in several iterations of the playtest rules.
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[2)] Slow natural healing. No HP are recovered after a long rest. Characters need to expend HD to recover HP.
Plus:
3) Second Wind grants temporary hit points. A character must be at or below half of their maximum hit points to use Second Wind.
Analysis
Healing: Combined with the standard rule that you regain half HD per long rest this means that a drained character will need 4 days without getting hurt to regain all of their hit points and HD, which is probably a good enough number for me.
Second Wind: Thp don't stack, so you can't use them to throw off the healing economy. Setting a below half hp restriction means you can't just automatically activate it and always start out your day effectively over your max hp, yet allows low level characters to sometimes end up with effective hp higher than their max hp. At higher levels Second Wind will be weakened if viewed alone, but its reliable presence in a game with weakened non-magical healing should balance it out. When combined with the healing rules above, the results would appear to keep Second Wind roughly as useful for both low level and high level fighters.
(I might also use lingering wounds, but am concerned it might slow play down too much, so we'll see).
Changing what the length of time is for a "rest" just to tweak healing will have TONS of unintended side effects.
Let's go with that gritty thing for example where a short rest is 8 hours and a long rest is a week.
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If you want to change healing rates, do NOT redefine the lengths of "rests". Instead, just move healing off of the "rests" system and give them times (hours, days, etc). Doing it like that, you'll avoid screwing up everything else.
Yes. A lot of people don't seem to get this. Amazingly enough, most of the same people who are attached to non-magical healing being slower and inferior to magical healing are the same people attached to other classic D&Disms like spellcasters using daily (rather than weekly) resources.*
* OD&D may have done it differently, but most old-schoolers didn't start in the paleolithic era (congratulations to those who got in on the ground floor though!

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