May I introduce you to erlang/BEAM? These issues were solved 30 years ago, and the ecosystem has had plenty of time to solidify best practices around exactly what you're asking for.
Default BEAM timeout is usually 5s (probably too long, in some cases), if you miss it the default is unhandled exception, which crashes the process that made the call (and only that process, no others). The VM will then recover all of the resources (file descriptors, sockets, data to be GC'd) associated with that process. All in zero lines of code.
Also you can have millions of processes per core, with minimal performance regression, do you're likely to notice it in monitoring before it becomes a problem.
To be honest I use elixir. Docs are better anyways. For timeouts that information is in the standard library, for example
https ://hexdocs.pm/elixir/GenServer.html#call/3
Note that call is even more sophisticated, it incorporates a liveness check on its counterparty to quit out before timeout if there's been a catastrophe. Note this is effectively a simple form of backpressure management that degrades availability gracefully under stress in favor of ensuring the integrity of existing connections. Because fallibility is so baked into the runtime, every good library incorporates timeouts where it's sensible.
For connection pools, it's not explicitly part of the standard library but basically everyone (maybe not whatsapp) uses poolboy:
