Does macOS support Ethernet over USB without a 3rd-party driver?
Yes - the vast majority of ethernet adapters work - especially USB-C since they are generally very new, a little more pricey and use fairly standard types of control chips and usb chips.
The Belkin you listed is both plug and play when the OS is booted, it also works at netboot from the hardware / EFI / firmware so it's truly driverless on the new Mac.

https://www.apple.com/shop/product/HJKF2ZM/A/belkin-usb-c-to-gigabit-ethernet-adapter

I personally am liking the Belkin adapters and all the thunderbolt 3 dock stations from the likes of OWC / Belkin / Elgato as well. Tripp lite and some others are on the market - but I've not had great durability of Tripp in a work setting, so the cost savings there aren't panning out over time over the higher end products for us. Light home use might be perfectly fine with less pricey adapters so YMMV.
Also - a general guideline if Apple is selling the adapters in their stores - they are likely best of the best. If they are selling them online, chances are they are best of the best or at least very good fit and need no drivers either.