Do you recall how the external drive is formatted? Also, is it formatted Mac APFS or HFS+ with Apple Disk Utility, or are you using "tools" that came with the drive or were downloaded from the manufacturer? If you are using third party drive utilities (which I and others i these forums generally recommend not doing) they could have become out of date with a recent MacOS update and hence stopped working to interact with the drive.
Try booting in Safe Mode, holding down the Shift key (this can take quite a few minutes) until the login screen appears. Does the disk show up now?
Try logging in as a different user, can the drive be seen then? Create a new admin user if necessary to try this. If it can be seen, then something in your original user account has changed or has something installed that is incompatible and that would need to be troubleshot.
When you are in the state where you do not see the drive in Disk Utility but can see that there is a device connected via System Profiler, can you find a process running in Activity Monitor called "fsck_xxxx" where xxxx could be HFS or APFS or something relating to the disk format? Does that fsck process keep running for a while? If it is running, it is automatically trying to repair a disk that it detects a problem with, this could take some hours, and when that occurs the drive is unmounted and might not be visible; you can also choose to kill that process manually in which case the drive should mount in Disk Utility immediately (and on the desktop), but possibly in an uncertain or degraded state. But you could then elect to perform repairs manually with Disk Utility or some other program. Suggest you consider DriveDx for hardware diagnostics only, run the extended (long) test on the problematic drive. Disk Drill or similar recovery software might be able to recover the drive if it is damaged.
Alternatively, try booting into Recovery mode with COMMAND-R held while booting. Then try to examine and repair the external drive with Disk Utility from Recovery.
Have you tried different ports and cables? Also, performing an SMC reset might clear an issue with the ports, although I think the likelihood of that helping is remote, but certainly harmless to try.
Can the drive be detected on another computer?