Improve event re-emitter logic across codebase

### Your use case

#### What would you like to do?
See https://github.com/matrix-org/matrix-js-sdk/pull/3796

We should examine if this sort of issue happens in other parts of the codebase.

#### Why would you like to do it?
Lots of potential to cause bugs.
We should have a better way of listening to changes on objects that isn't based on manually setting up re-emitter logic.

#### How would you like to achieve it?
Requires deliberation, but observables (eg: https://rxjs.dev/) may make sense. For example, we could have an observable map of users that the memberlist would subscribe to.

### Have you considered any alternatives?

_No response_

### Additional context

_No response_

Review pull request 3796 and inspect the memberlist's proposed observable map of users as entry points. Identify other manually maintained event re-emitter logic across the codebase, then define and agree on a replacement approach and its scope; the issue does not name specific files or tests, so completion criteria need to be established first.

Written by the indexing model from the issue text.