The beauty if the original question, is that there is no right or wrong answer, only personal opinions.
For example, if a dog gets hurt so bad in the field that it may need to be put down, I'm the only one that's making that call and I can't make it if my dog is there without me.
The friend of mine that is hunting two of my dogs right now, also happens to be the man I bought both dogs from. With the pup, I told him to cull as he would his own...

I figured he bred her, and he knows what his expectations are of his dogs. If she does not fit the bill of what he was looking for in that breeding, then do what needs doing. That same person went hunting with 5 dogs yesterday, and came home with three.

My pup made the cut yesterday.
I would also trust him to make the right call about when to hit the vet, vs when to put one down if needed.
It's not for everyone, but it works for me. BTW, he is about to ship 15k in heifers for me to use team sorting... If he can trust me with 15k in his cattle, I can trust him with a couple dogs.