Seems to me that you are doing everything correctly and the issue is software related.

I bought an HDMI to USB3 adapter, but the problem with this (and almost any adapter of this type) adapter is that it has no drivers for Debian.

there is your problem. It has no drivers for Debian. No drivers means never gonna work.lol. Your adapter is the issue.

Try this docking station from Dell. There are 2 of them, one supports 4k and the other doesn't. Make your pick and give it a shot.

I got that Dell docking station link from here.

Moral of the story. You'll need to find an adapter whose manufacturer chooses to support linux. Hope you find this helpful.