There is a PR for Wayland support, but there is a mix of things that do and don't work. If you're interested in contributing:
https://github.com/godotengine/godot/pull/57025
You'll have to compile it if you want to try it out.
For your other question:
Aside from requiring less maintenance in the future, one of the inherent advantages of Wayland is that it reduces input latency by cutting out the middle man.
With Xorg, programs tell the X server that they want a window, and then the X server talks to your compositor to get one. Rendering and input must pass through the server.
Under Wayland, programs and compositors can communicate directly without going through a server, therefore reducing latency between an input and result.
Although "DisplayServer" is used in discussions, Wayland is display-server-less. It is only a protocol to make sure all wayland programs and wayland compositors play nice with each other.
The only thing that really holds back Wayland IMO is the lack of interest to push support. Xorg is already stable and works well. But distros that support Wayland by default are few by comparison, and a number of popular programs still don't support Wayland natively.
Furthermore, if you're an Nvidia user then you're going to have issues due to their closed-source drivers, and Nvidia users are a massive portion of the market.
As an Nvidia user having tried GNOME and KDE's wayland implementations, it is not bug-free and you will absolutely encounter issues in its current state. Less so on GNOME than KDE, but they are still livable. And even though OBS officially supports wayland, it doesn't work on Nvidia.
I'm planning to get an AMD GPU once prices drop since most of my issues are apparently specific to Nvidia's drivers.