Are you changing non-rest-based healing (spells and similar) to adjust for the increased deadliness of being dropped? Currently a healer burning all of their slots can't generally match the incoming damage of most encounters. With these rules, there will be a lot more incentive to just nova as hard as you can at the beginning of a fight to reduce the number of combatants and therefore incoming damage unless using those resources to heal during the fight is made more realistic. The reduces number of encounters per day will also encourage, and increase the ability to nova for those classes with repeated nova capability.
As a side note, in my games, I've trained the players into leaning heavily into using HD between combats and rarely use healing spells or healing potions (healing potions are extremely rare in my games, and the party sees them as real treasures). I haven't quite figured out what to do with Paladin lay on hands (considering making it 2 hp/level healing), but then again no one tends to play paladins in my game for some odd reason.
The cleric & druid players enjoy this because they get to use their more interesting spells instead of having to load up on every type of healing spell that comes along.
As far as dropping to 0 HP in a fight, for my own games I'm not so worried about that initial drop to 0. However, I keep a table for when that first death save is failed for "lingering injuries". It consists of scars, broken bones, torn muscles and other injuries that potions and cure magic simply can't fix. They all have a progression for a subsequent second failed death save - for example, the broken leg is actually also crushed and even if it heals you may always have a limp. Get to 3 failed death saves but get brought back
revivify*? Well, you may be alive once again, but it may be that leg is gone...
This also has consequences if the PC happens to fixed back up and drop again. Could be the first time it was their leg. Second time maybe their sword arm. A third? Possibly a concussion. It also gives some reasons why old war veterans and adventurers might be missing an eye, walk with a limp or sport other old wounds that are a testimony to such a dangerous life.
* I also have a mundane system that allows those who have failed 3 death saves to be "brought back", but usually at the cost of a permanent scar or other side effect at their brush with death. Though just as many times I'd have players prefer to stat up a new character rather than bring the old back to life.