I don't think mining is the best application of utilizing client side connections. But surely we can find profitable ways to distribute computation. I'd be willing to divert the resources of a small sandboxed vm of sorts to websites if it meant less/no ads. It'd be fascinating to see what people could come up with if there were a publicly distributed Lambda/Serverless sort of capability.

I don't think any user would want to, but some unscrupulous websites and ads already do.