The original post is really funny to me because my 6yo cousin did almost that exact thing while playing TTR with us at Thanksgiving. Except she collected all cards, not just the rainbow ones. She was good about taking turns, though, so whatever floats her boat.
For your son, have you considered games with more of an active/hands-on element? I remember playing "Mousetrap" with friends, and we had LEGO Creator, which is now outdated but it looks like they have a lot of different LEGO games now. I never got really into these two but I was a bit older when we got them. I also remember being immensely jealous of anyone who had the "Go Fishing" game (with the magnetic fishes and the tiny fishing rods and the rotating carousel). I also recall my sisters tended to like playing games of chance, at least with me/adults (Sorry!, Chutes and ladders, that kind of thing). Ticket to Ride is great but does involve a lot of strategy and might put the 5yo at a (perceived, at least) disadvantage.