Integrate simulated time with testing/synctest

Go 1.24 includes an experimental testing/synctest for controlling time.

Since we need to control, at least, the passing of time, it should be worth considering, if it could provide value to utilize that package; to keep simulated time on server and frontend in sync.

Although I'm sceptical if it brings real value in tests.

For the server, it should mostly be about timestamps recorded on entities, that can be verified they are presented correctly in the browser.

For client tests, it's about testing throttling/debounce behaviour.

So there probably is little reason to keep the two in sync.

Start by evaluating Go 1.24's experimental testing/synctest in the context of this repository's server and client tests. Determine whether it provides value for entity timestamps, browser presentation, or throttling and debounce behavior, and define a focused scope before making changes.

Written by the indexing model from the issue text.