Does macOS support Ethernet over USB without a 3rd-party driver? As we all know, the MacBook and MacBook Pro do not have Ethernet ports.

Apple makes a Thunderbolt Ethernet adapter (which requires no driver), but with the change to USB Type-C, this is no longer an option without an additional adapter.

Now Apple sells a USB-C Ethernet adapter made by Belkin, but according to the reviews it appears it requires a 3rd-party driver. I've also looked around on Google about this and other adapters, and I'm getting conflicting information on support for Ethernet over USB without 3rd-party drivers.

Does macOS have the ability to support USB Ethernet adapters without 3rd-party drivers, or is searching for one a wild goose chase?
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.