The issue is that short rests do a lot more than recover hit points in 5e. Warlocks, Monks, and Fighters in particular heavily rely on short rests being a regular part of a balanced adventuring day. That kind of makes them sound like a breakfast cereal, but it's still true.
I don't think it's going to encourage a gritty style. As written, you're just encouraging long rests in favor of short rests. You're telling the PCs to stop adventuring when they're at half hit points because non-magical recovery isn't possible. It's not going to make the game gritty. It's just going to make the PCs extremely conservative. You won't have more PCs dropping to 0 hit points, you'll have PCs stopping to long rest after dropping below half hit points. If you want to make it grittier, then I would change the timing of long and short rests entirely.