At the moment there is not much use for the first aid skill. Investing into that skill as a main char is rather a waste of trait points since most of us try to avoid getting injured by zombies anyways, and you either get lucky scratched or lacerated and survive or you get bitten Z- infected ... and simply die anyways so first aid becomes pointless then.
And this is where I would like to see more situations were you can get "domestic injuries" that don't seal a death sentence like a zombie bite. I know you can get cut damage from broken glass and even running through bushes can give you scratches, but I would love to see (and that may be optional in the sandbox settings) every day household injuries due to low experience in any field.
-You are bad at cooking? How about you cut your finger when cutting vegetables or a dead animal, or you burn your finger when cooking / touching a hot pot.
-If you are bad a carpentry, there could be a chance that you hit your own hand/finger with a hammer when building something.
-Get a splinter under your skin (use tweezers) from sawing logs.
-bad at tailoring? prick your finger on the needle
... and so on.
It could also be countered/mitigated by wearing protective gloves/clothing.
The chances for such injuries should be fairly low, like 0.1 - 5% based on your respective skill, or completely optional.
Situations like these would make the first aid skill a lot more needed overall while adding to the immersion, without an immediate life or death situation from a zombie injury.