Earlier quoted context omitted.
UDP hole punching does not work with all types of NAT, e.g. symmetric NAT where a user can appear from the outside to be coming from multiple different IP addresses. (This makes the STUN server useless for telling the other peer where to connect.) Symmetric NAT is common in corporate networks, which I suppose is not the first place you'd find gamers (or you could just run LAN games with co-workers). I guess WebRTC is…
"UDP hole punching does not work with all types of NAT..." I think what you mean to say is the STUN and TURN solutions do not work. What else have you tried? I personally do not use those solutions. The last resort is for a peer outside the problem NAT to forward traffic. This is not true peer-to-peer (IMO) but it does work. If what you suggest were true, that reliable peer-to-peer is "impossible" because of some typ…
Ideally it would not be necessary to try anything else and peers would just be able to directly connect. In practice TURN is a standardised way of doing relay, and what else are you going to do? Invent your own? How will you test it across the 1000s of different network configurations and know with any confidence that it actually works across the open Internet? Skype might have the resources to pull that off, but the next peer-to-peer startup might not.