As for the NAT problem, that's mainly an issue for peer-to-peer scenarios. If you have a publicly addressable server at one end, you don't need all of the complications of a full ICE stack, even for WebRTC. For cases where you do need TURN (e.g. for WebRTC with clients that may be on networks where UDP is completely blocked), you can use hosted services, see https://iceperf.com/ for some options.
And as for MoQ - the main thing it requires from browsers is a WebTransport implementation. Chrome and Firefox already have support and Safari has started shipping an early version behind a feature flag. To make everything "just work" we'll need to finish some "streaming format" standards, but the good news is that you don't need to wait for that to be standardized if you control the original publisher and the end subscriber - you can make up your own and the fan out infrastructure in the middle (like the MoQ relay network we've deployed) doesn't care at all what you do at that layer.