If I do a firmware update via
sudo rpi-update on Raspbian Buster on the Pi 4 and then flash another OS (e.g. LibreELEC) to the SD card, will the firmware still be up-to-date?
rpi-update command updates the kernel. The SD card's boot partition holds the kernel. So if you re-flash your SD card, the kernel will be replaced.
If you meant to say
rpi-eeprom-update though, that is different. This command updates the RPi's internal EEPROM storage, and it will not be reset if you re-flash the SD card.
BTW you should never have to run either of those commands. Countless beginners have bricked their SD Cards with an unbootable experimental kernel, and the EEPROM is kept updated automatically via APT.