Thanks for all the suggestions but none of them were able to solve my case. I came across ThatOneComputerGuy who suggested disabling and enabling Bluetooth and WiFI through the UEFI BIOS:
- Boot into UEFI BIOS through "Advanced startup"
- Disable Bluetooth and WiFi
- Boot back into Windows
- Go back to UEFI BIOS again
- Re-enable Bluetooth and WiFI
- Boot back into Windows.
This made it work for my case. I had tried different driver versions, downloaded the Surface firmware and drivers without it solving it. This did the trick for me. I guess this is a more reliable way of hard-restarting your network card without physical access to it, or betting on it will fix itself over a "good" night sleep.