Are you going to go looking for a 3 legged squirrel if you misjudge the distance? i dont think so.

If I have to yes no matter what pain or time it takes. I'll search for hours for rats that other people have hit that haven't died, let alone if I hit one that doesn't die straight away.
At that sort of range if I judged it wrong I would miss either way, any form of hit from such a fast round that expands the way federals do, is going to remove so much body matter it would die instantly.
For the record I hit the squirrel in the chest. There was a small slither of skin holding it's back legs to what was left of the head, the rest of it was spread out behind it. Not a nice icture but that was the way it was. Very dead and in no way did it know I was there or there was a bullet coming. In my view that is a stress free way to die.
As for seeing them I use binoculars alot. With respect to your air -rifle I'm in the same boat as you. I can hit well to 50yds with it but shoot no more than 35yds. If I know the ranges are likely to be longer I use the rimmy. If it's not safe for a .22, I build a hide as close as I can and shoot from there.
I spend alot of time researching and practising. I know I can shoot well to 300yds, but I don't generally tend to. The furthest fox I've shot is around 200yds. The shot felt good, the bullet was about 3/4inch away from where I wanted it. If I hada deer legal gun would I take 200yd shots, no I wouldn't. Maybe after alot of training and researching into the effects of certain bullets on deer at various ranges, I may start practising out to that range, but it would be a long time before taking the shot at a living target. Plus the other side of it, I can't drag a deer, or carry one, and I certainly aint leaving the carcass where it drops. Hence why I don't shoot them. I may start shooting muntjac soon. but only when I've researched different bullets and know that I can call on someone to help me with the carcass.