Due to the dissatisfaction with the Hit Point bloat of 5e and the forced High Magic Setting (Cantrips, Potions of Healing, so many Spellcasting Classes...etc) ...
I'm thinking it might provide that grittiness our group desires and that we have been missing with the last few editions.
You should check out Hackmaster 5e. It's gritty, low magic (still has spells), and the combat is a cool count up by second every person acts every second kind of thing. It reminds me of a polished 2e with new bells and whistles.
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The solutions for more "realistic" (ha!) healing are in DMG: (i) slow healing (ie you dont get HP on a long rest, just half your HD back, which you can then spend for HP) and (ii) Injuries table activating off being reduced to zero hp. We have an expanded "Injuries and setbacks" table to specifically discourage the "yoyo" effect and make being reduced to zero hp more meaningful.
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In my campaign, PCs do not recuperate hit points during a night's sleep, only half their hit dice as per the normal rules. They can then use those hit dice to heal themselves as they see fit.
I use the slow healing rules with one modification. You get back all your HD. I've found it's a nice balance to keep the players moving forward while creating the "stress" of managing HD.
I am also using some house-rules regarding lasting injury. You can check them out (and my fear rules) on this shared
Google Drive.
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OP - There's a balance between the "realistic" feel of what we might consider a "real world" effect, and the practical need that the "show must go on!". I have run a menagerie of various systems from GURPS to every version of D&D to Unisystem to Hackmaster 5e to HARP to Hero and all kinds in between. They have a different feel certainly. One effect I noticed in "gritty" games is the trepidation of the players. They get cautious, overly so in my experience. I appreciate smart players! However, the game gets SUPER boring when people start debating how to assault the dungeon or how to sneak into the castle... Planning is good, but excessive planning bores most players. I feel that 5e strikes a good balance.
In my game, the characters have faced some disturbing and strong opposition in the tunnels below the capital city. Several NPCs died in the last conflict. The PCs all lived, but they were down some hit points. They have access to magical healing and few potions discovered along the way. They are STILL cautious and wary of going back down there to "clean up" any creatures left behind. The stats on those creatures do not make them especially difficult for the 3rd level party. They do have PLENTY of help with an attachment of fighters from the church. My descriptions, the chaos I introduce in the battle scenes, and the creepiness of the location have all supported the feel I was hoping for. The HP hardly matter. I painted a scary scene and the players felt like they were part of a scary scene!
As I'm running 5e I am reminded of the wonder I had in older editions while being impressed with the new design model. Things just work. It runs smoothly, quickly and most importantly we are all having fun. I came back to D&D after a long dance with more complicated systems looking for "that feeling" many of us system-wanderers are looking for. I grew very tired of all the tracking, fiddling and complexity in game play. It all got in the way! Now we just play, and frankly there is no substitute for it. Playing is the fun part of TTRPGs!!
My advice - don't sweat it. You can slow down HP recovery with the simple options in the DMG without seriously hampering play. The game is designed VERY WELL. I say this as a 33 year veteran in this hobby having played a wide variety of RPGs. Focus on the adventure. Let them explore and see where the dice fall. Adjust as you go along. Focus on the fun parts and you will find your players are having fun!
Just my two cents...