I bought a UGreen USB 3.0 adapter and it arrived today. When I took out my Switch from the dock after a couple of hours, I noticed it being slightly warm (which I never noticed before).
I unplugged the ethernet adapter and was going to check tomorrow if the Switch was cold after that, but decided to search for it online as well... I guess I know the answer before I check tomorrow.
I'm glad I noticed it after half a day, but it sucks that it doesn't work properly (even with the licensed Hori adapter). I'm on 6.2.0 since yesterday, so that hasn't fixed it either.
I guess I will just keep using wireless then. I bought the ethernet adapter because wireless reception can be spotty where my Switch is docked (it's fine with the beefy antennas of my router, but not with most handheld devices). Usually it works fine (although somewhat slow to download software), but other times I can't even browse the store properly. I guess I will keep living with that. I should have checked online for problems with the ethernet adapter before I bought it.
Some more info:
I let all the DNS requests run through a pihole and there aren't more requests when it's running on the USB ethernet adapter compared to the wireless adapter (both adapters have different IPs so it was easy to check). It just seems that running a USB device warms up the switch for some reason. The weird thing is that (from what I read) it wasn't the case prior to the 5.0 update. Not sure what they have changed.
Edit:
It clearly makes a difference in temperature for the system when I unplugged the USB ethernet adapter to when it was running.
When the USB ethernet adapter was plugged in, the Switch was noticeably warm to the touch (it didn't feel hot, but it was clear something used energy). The Switch felt cold after getting it out of the dock with only wireless running.