Cats - like humans - have their own personality. Some love to cuddle and are very social, others don't like to be touched. This cat seems to be the latter.
You should be aware that not every cat needs to be inside a house and sleeping on a bed to be happy. You should not try to impose your idea of a "perfect cat life" on him. You can adopt him and care for him while still giving him the freedom to come and leave whenever he wants.
You should also be aware that male cats are less social than female cats. We have a lot of questions on this side of cat brothers chasing away their sisters or a new male cat being aggressive towards other cats living in the same household. All your cats will need a long time and a gradual introduction towards each other.
That said, the best way to catch him to be examined by a vet might not be to grab him and push him into a transport box, but to let him walk into the box himself.
If you already have a suitable transport box, simply put it into your backyard near the place where he likes to be. Let him get acquainted with it on his own terms and in his own time. To aid your efforts, you can put little morsels of foot inside to lure him in the transport box. Let him move freely in and out the box for several days before you try to trap him inside, or he'll avoid going inside after the first attempt.
If you want to touch him, use the same strategy. Sit calmly in your backyard and let him come to you. You can lure him with food, but you should give him time to trust you instead of pushing the matter.
First put a bit of food near you and let him eat it without interacting with him or looking at him. Then you can offer the food in your hand. Let him sniff you and eat the food from your hand without trying to touch him, or you might scare him away. Take the next step (light touches to the front or side of his chest, where he can see you and can move away) only several days after he willingly approaches you.
You should not look him directly into his eyes, because in cat language this is a warning signal. Do the "lazy eye blink" instead to communicate that you are no danger to him. You can verbally greet him with his name to show him that you are aware of and accept his presence (as opposed to sneaking up to him or chasing him away).