Are both Macs running the same version of macOS including the same patches?
It is possible macOS Catalina or Big Sur is not compatible with your adapter and/or monitor (or maybe just a recent update made them incompatible). I suggest booting to an older version of macOS to test this theory although if a recent firmware update is the problem instead of a software issue, then even an older version of macOS won't help. Apple has not had a great track record with macOS updates for the past couple of years.
If you don't have an older version of macOS on a bootable external drive, then you can boot into Internet Recovery Mode using Command + Shift + Option + R to access the online installer for the OS which originally shipped on this laptop from the factory. Or you can create a bootable macOS USB installer for macOS 10.11 to 10.14 using the instructions in this Apple article:
https://support.apple.com/en-us/HT201372
If you don't see any video on the external display while booting to the macOS installer, then close the lid of the laptop to force the video to the external display.