If it helps, Rayxuln's spine-module-for-godot (linked above) compiles for web with no issues (on my end, at least). I'll be using that while the official support spends some more time in the oven.
If there's one thing I really hope the official Godot support adds, it's an easier way to attach arbitrary Godot objects to Spine attachment slots. I mean, it works, I can use it, but it's really clumsy and unintuitive. (And undocumented, IIRC - I only figured out how it worked by looking at the code itself!)