Yep. If you didn't see already, bandages have also been suggested below.
It's also something I had thought about but ultimately decided against because I thought it would kill too quickly.
The bandages would be a good addition though, and help with the dying too quickly thing. And an option to turn off bleeding damage could be added.
Though I'm not sure how easy/possible it'll be to implement. (Just taking damage from bleeding should be easy but will require reworking some of my logic, but the bandage could be difficult/not really practical.)