Perhaps you would like to add the server https://variant.baduk.club/, as announced at https://www.reddit.com/r/baduk/s/E1pmX1hYwe to the list in your first post. And anyone following this thread but not r/baduk may be interested too.
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I actually tried their website. You cannot mix and match variants.
I see what you mean, I think. I just tried to specify Tetris on a custom board, and the preview suggested it was possible, but then a normal board appeared for the game
Ah bummer thats just a bug 
mix-and-match should be near parity with kahv server. Ill take a look tonight
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Actually “mix and match” suggests to me more than just allowing arbitrary combinations of boards and rules. They talk of “combine several compatible variants together to create something entirely new”, which sounds considerably harder to specify and implement! But I suppose it may be possible to factor out some other aspects such as stone visibility or turn alternation and allow the user to pick each.
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The first post is a wiki, so you should be able to add it yourself, if you don’t mind doing that.
Variant Go Server was way more focused on the “mix and match”, which was actually a problem for implementing some variants we wanted back in the day. We briefly considered contributing to it too. As some early posts of this thread show.
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Posted a PR at Fix board shape/variant config desync in game creation form by benjaminpjones · Pull Request #515 · govariantsteam/govariants · GitHub!
But I suppose it may be possible to factor out some other aspects such as stone visibility or turn alternation and allow the user to pick each.
Yes, we absolutely have a pattern for composing variants! A good example is Rengo, which can be applied to any variant (props to @martin3141 for introducing this!). I believe it would be relatively straightforward take the same approach with other variants like phantom, one-color or thue-morse. It’s not something we’ve prioritized, but I don’t think it’s a huge lift and I can see the appeal 
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