Clock doesn’t stop when the game closes in any speedrun category that would do it on purpose.

Yes, it does. Souls speedruns are timed by in-game time and in-game time pauses when you are outside of gameplay. Doesn't matter if it is a force quit out or a normal quit out or even a loading screen, the timer pauses.

Speedrunners show the IGT at the end of the run and they use SoulSplitter a LiveSplit plugin.

Here is DS1 Any% Force Quit World Record and, as you can see, the timer pauses.

Here is the Former Zips World Record in Elden Ring and the timer pauses during load screens and quit-outs.

Might be difficult for streaming/live events if it messes with the OBS setup. Having to reconfigure after relaunching takes more time, too.

They usually capture the desktop in OBS instead of the game window (you can see the desktop in the DS1 run). Runs with Force-quits have also been done at GDQ with little problems.