Found a disabled stray kitten.. what to do?

My mother lives in a kind of dodgy area in a big city and there are always stray dogs and cats running around. Yesterday there was a cute little black and white kitten hanging out under my moms house. He is very skinny and dirty but also very sweet and affectionate. He also has a bad front leg. At first I thought maybe it was broken but when I started moving it around and feeling it I noticed he didn’t have any pain. In fact, he didn’t seem to have any sensation on it what so ever. He is basically just dragging this arm around but he has absolutely no use for it.

I plan on taking him to the vet to get a check up and all his vaccines but I don’t have any money to get x rays or anything major. My parents don’t mind keeping him because they are animal lovers and don’t have the heart to turn him into the shelter because he’ll probably get put down because of his disability. The only problem is if he stays with my parents he wouldn’t be exclusively an indoor cat.

So my question is, what are the chances of this kitten living a happy, normal life with this defect? Should I try to re home him or should I just turn him in to the shelter?