Since you are of of the people working directly on this codebase, may I ask you why is select! being used/allowed in the first place?
Its footgun-y nature has been known for years (IIRC even the first version of the tokio documentation warned against that) and as such I don't really understand why people are still using it. (For context I was the lead of a Rust team working on a pretty complex async networking program and we had banned select! very early in the project and never regretted this decision once).
What to use instead?
Your favorite llm will give you several good options. You can use an mpsc channel where you have a task per sender which waits on each branch in the select and then sends a message and then just wait on the receiver end of that channel. Or you could use https://docs.rs/futures/latest/futures/future/fn.select.html (or the select_all version). Both make it more obvious what is going on. The last way would be to implement a future manually. This would probably be my favorite option in many cases as it would be least overhead, but if you've never implemented a future it may be a bit intimidating.
Edit: tokio docs actually provide several suggestions: https://docs.rs/tokio/latest/tokio/macro.select.html#alterna...