Not enough information and kind of confusing. I own M1 MBP for 4+ years and did not need to boot into Recovery even once. You can create Volumes and mount external Time machine from admin account and with admin password from regular account. Why using Recovery mode at all? Booting to recovery should be kind of last resort fixing tool - so what went wrong before?
For Time Machine external drive... It occasionally disappears. Especially if there are other issues with drives and OS is trying to mount other drives unsuccessfully - or, sometimes I suspect, simply because it can...
If you unplug and plug again, does it mount? If you restart the system and external drive, does it mount? If you go to Disk Utility, does it show greyed out and can you right click it and select mount?
Does it mount on another computer?
One needs to isolate if the problem is OS, M4 hardware, or external drive formatting or hardware to know what one is actually fixing.
