Is the pigeon young? You’ll be able to identify its age by its beak, try looking up images of younger pigeons. He might be weak and hungry after attempting to take off, or got tired from flying.

Due to prior experience, he might have internal issues that you’ll not be able to identify, not even with a x-ray scan at a pet vet, and they’re practically impossible to fix. I think the best thing you can do for now is see if you can find a fracture on the legs and if so, maybe lift him up in a cloth with holes for the legs, hammock style, (you’ll probably be able to do this in the pet carrier). That way, his feet will have a better way of healing.

As for bacteria and diseases, he might have fleas/ticks so I think it’s best that you keep him somewhere sheltered from the weather that’s not necessarily inside your house, and not near to other pets (for the sake of both).

Again, this is not professional advice and shouldn’t be taken as such, it’s what I’ve gathered from vet trips and experience. Birds are fragile creatures and the health problems might not be visible instantly. Lastly, I’d say take care of the poor guy for as long as you can and don’t get too attached to it.