Elixir doesn't affect most of the operational aspects of Erlang, and it certainly in no way affects performance. You're still running on the BEAM, except now you actually have more apps running (The Elixir runtime runs as an app).
Syntactically it cleans some things up, adds some niceties, and annoyingly makes atoms require a prefix of ":" (annoying because atoms are central to Erlang's readability).
Personally, I don't find the changes to make a big difference when using the BEAM, and I prefer the familiarity of Erlang's syntax. It's also the language all of the documentation for the BEAM will use.
If you absolutely can't get your coworkers to ditch dynamic/gradual typing, Elixir or Erlang are still great choices and way better than Python/Ruby. Otherwise, try Rust.
Are you actually using Rust for professional web development? I've tried it earlier this year and my experience wasn't that great, so I'm quite surprised to read this comment.
What issues did you have? We haven't had very many issues. We rebuilt some internals tools, but for the most part it's been smooth sailing. We do use hyper and h2.