No, it does (did) not.
That's why I thought that probably the camera itself was broken. That's why I tried several MacBooks and they had the same problem with the built in Thunderbolt camera. But then I found out that it worked well in Bootcamp Windows, both the camera function and the LED, so I concluded it must be a driver/OS problem.
The LED did light up every time I booted into Catalina but extinguished again within a fraction of a second. No camera was listed in the system overview. So, it behaved like some driver crashing every time.
But, suddenly, this morning (European timezone) when I switched on my Mac again I decided to have another look in system overview first. The camera was listed and remained on the list, so I tried a few apps and the camera seems alive and kicking again. There were no upgrades in between, I did not change anything myself, just booted up normally like every morning and suddenly it's there.
Strange, but it seems to work for now, and continuous to do so even after several reboots.