Can confirm that is happening to me too, using Firefox v130 on Linux, the weird thing is I have another machine that was logged in since a few months back that is still logged in and does not have this issue

You can do a local recording with zoom, you can have a meeting by yourself, share screen, and then position the camera where you want it, and record. when you finish the recording and the meeting, the file will be processed and saved on a local folder without doing anything else.

I consider that the easiest method.