My wife has been experiencing this issue and i'm the geek of the family so have been trying to figure this out for her. We have identical Macs and on mine, with no external drive, i have no problem. On hers, I've tried various configurations and all the troubleshooting steps and it seems clear that if I modify the Libary Media storage location to the System drive, it works but if the Media storage location is on an external drive, it does not.
Answers to your questions:
FCP version? 11.1.1
MacOS version? Sequoia 15.5
Which Mac? MBP 16 inch Nov 24 M4 Max
Does FCP have Full Disk Access enabled in System Settings? yes
Are you using external storage for the library? If so, how is it formatted? APFS, MacOS Extended or other? Yes, APFS
How much free space on the System drive? 560GB
How much free space on the external drive if one is being used? 1.47TB
Have you tried resetting FCP preferences? Hold option+command while launching. Yes
Have you tried with the built-in mic while the external mic and interface are disconnected? Yes