I ran two games of Curse of Strahd which ended recently, and ran a much more "gritty" (for lack of a better term) game. Here's some of the things I had:
Healing Word and Cure Wounds did not exist in Barovia.
A Short Rest was 8 hours, a Long Rest was 7 days.
The '3 Death Saves' system was replaced with the Exhaustion chart, and Lingering Injuries occurred when PCs reached Exhaustion Level 4.
I did allow some "healing" to occur... players could still spend Hit Dice, the Healer feat could be taken and used, Paladins could still Lay on Hands, and Potions of Healing existed and herbalists could still create them.
The results were as follows:
A 7-day Long Rest period was too long. The actions of the PCs in both groups in Barovia could not warrant "taking a week off" from both their actions and the results of their actions. Time could not "stand still" as it were to let them regain their health because the plots and stories they got themselves involved in would not let them. As a result, I ended up having to shorten Long Rests down to about 3 days (+ a day or two if a PC was in Exhaustion Levels 4 or below) just to allow forward momentum of the story while still trying to maintain some semblance of "grittiness".
What this told me is that it would be impossible for me to ever create a "realistic" type of recovery via rest situation in D&D, unless I purposely created short adventures that allowed for Downtime between them (and thus could also lay up wounded PCs). Curse of Strahd was definitely not that kind of adventure.
An interesting alternative is that encounter difficulty should generally be much lower if you use slow healing. If getting wounded with several levels of exhaustion at 10% of your HP takes a week recovery, that's essentially a "failing" scenario. With this in mind, long rests should be seen as more appropriate for between adventures, and any time-sensitive adventure should be able to be completed without a long rest. It would probably take some adjustment, but once the players understand that even minor injuries can lead to them failing, then much less deadly encounters become much more significant.
Since that is so different than typical play, I guess it's an open question whether the players would ever actually adjust.