Then it is only Opera that is incompatible (as it is based on Webkit like Safari).
I can just guess that it was implemented together with introducing GLES3 for Godot 3. But later given up because to many problems arose and WebGL2 will anyway be removed from Godot 4 because OpenGL3 will be replaced by Vulkan.
I propose that you create your project in OpenGL2/WebGL1 (if you decide to continue to use Godot) and then move up to the Vulkan based browser support when it gets available some time after Godot 4 is released.
Although there's no fixed date for Godot 4 release and especially not for some WebGPU Export template.
That aside, Godot is open source. If you don't want to miss WebGL2 then you could debug & update that Godot export template yourself and contribute it to the community.