Hello,
I know it has been stated already that logical games bots are not going to make it into the web version of Playroom.
though, that was back in times, when web clients couldn't handle computationally difficult tasks very well, and thus the heavy lifting was left on servers, that (quite understandably) couldn't handle all the players.

But times have changed, and recently, the WASM technology of client-side running native code on web is getting more and more popular. That applies for chess engines as well:
https://github.com/ianfab/Fairy-Stockfish

Fairy stockfish seems to be able to play a ton of games, including (from our list) chess, Reversy, Othello and Connect-four.
And there is a lot more, support could easily be added for Chess960, Crazyhouse, Shogi, Minishogi etc.
Since the engine is available both as a standard binary as well as WASM, it could be used in the web client as well as the native client, unifying the used techniques.

The only problematic part I can see is the license. Just like Stockfish, FS is also licensed under the GPL, which is a known troublemaker when it comes to including to non-GPL projects.

However:

1. While using a WASM library is indeed a somewhat tighter link than subprocessing, which you normally do with native Stockfish, at the same time, it's not like you'd be using some special API of the library. Even if it's WASM, you're still communicating using the standardized UCI protocol, and could easily swap FairyStockfish say for the pure Stockfish, or for a WASM Shogi engine. But then, what's really the difference from subprocessing? If you really wanted to, you could make a GPL middle layer called subprocessor, that would emulate the subprocessing functionality say via websockets. So it's very debatable whether there would be any violations of the GPL at all.
2. Even if you actually didn't follow the license so tightly, I highly doubt Stockfish developers would care a slightest bit. Not in a project like this. There was the sound sue of Chessbase, but Chessbase is a well-established commercial company, that literally stole parts of the project and advertised them as their own awesome invention, unique on the market.
On Playroom, it's not a secret Stockfish is being used, you're earning nothing from the project but bills, so I can't really imagine someone having problems with it. WASM is the invention of the last years, stockfish developers couldn't possibly have predicted it will be so important in the future, they have chosen the license knowing closed programs will be able to use it, so from the idea's point of view, WASM should be no different, even if the license is a bit unfortunate in this regard.
The only-one I'm unsure about is the FS developer, since he made some considerable expansions of the project. There is not much that could tell us his attitude, he had no option but license the project as GPL, so this was not a deliberate choice on his end.
Though, he's quite active, so asking him about his opinion shouldn't be a problem I think.

Either way, I just wanted to put this up as a suggestion. Be it Stockfish, FS or another engine, WASM seems very promising, so I think at some point, web bots on Playroom wouldn't necessarily need to be such an unreal idea.