To be clear, I'm not talking about the server implementation only about implementation on the browser.
And of course, Rust and C are faster than javascript, and it will be noticeable on search algorithms, but IMO, it will not cause user loss. It is smarter to get to the stage you have those users as fast as you can to validate this need.
So I would not go into implementing new AI libs in rust as part of a turn-based game engine just for the performance gain--mainly because it is a tougher project than building a turn-based game engine!
I would search for existing, viable solutions. As I said before, this includes server AI and server trained models running on something like tensorflow.js. Additionally, I will also try to research browser libs that may use compiled webassembly and webGL. In any case, I think it's not wise to build your own for this purpose.