the Fireface USB Settings app can be opened, but Fireface UCX II is not shown in macOS's Sound Settings, thus I cannot select the device and cannot use it.

Not in Audio MIDI Setup either.

You got a manual, the latest version online available here, in case you never used it yet:
https://www.rme-audio.de/downloads/fface_ucx2_d.pdf
https://www.rme-audio.de/downloads/fface_ucx2_e.pdf

Your PDF viewer has a search function, search for Class Compliant or CC and then you find:

Manual ch 32: "The Class Compliant mode can be activated and deactivated directly at the unit, as long as it is not connected to a Mac or PC: SETUP/REV, Options, Hardware/Diagnosis, CC Mode On."

Unplug USB cable and then you can change it.

But as mentioned already, you only need CC mode for iPAD, for Windows and Apple you have/need drivers to get all the functions and TM FX.  Only use it for testing on Windows, Apple, iPAD to see, whether it will be recognized over USB in that driverless mode.

"Neither Windows nor Mac Class Compliant support is relevant, though, since there are dedicated and matured drivers for both OS X and Windows, which provide ALL the device's features at lowest latencies. The main reason for implementing Class Compliant mode was to allow connecting the UCX II to an Apple iPad®!"

Besides CC mode it also mentions additional important installation requirements and points to another document.

Chapter 13.1
"During driver installation the programs Totalmix (TotalMix FX) and Fireface USB Settings are copied to the Applications folder.
They will automatically start into the dock if a Fireface UCX II is connected. A reboot of the computer is not required.
Driver Updates do not require to remove the existing drivers. Simply install the new driver over the existing one."

and

"For the latest M machines (Apple Silicon) please read the included readme carefully, as on such machines the driver installation requires additional steps to enable third party kernel extensions."

Read my previous posting and the manual. Updated my posting and gave you concrete hints.

I was in a hurry before, now with more time why I suggested to try the cc mode: simply to find out if the issue is hardware related (faulty USB port on UCX II -> cc mode wouldn’t work) or software related. So it’s a step forward to see the unit working and being recognised in cc mode.

@Faust8329, as you can read in the manual, in cc mode there is no Totalmix.

Concerning reduced macOS security and kernel extensions: that’s irrelevant for the driver v4.09! This version is not a kernel extension. You can set your Mac back to the default settings.
But it’s important to do all the described steps after the installation (enabling 3rd party drivers in system settings -> privacy and security).

After doing all the above, I am still not seeing Fireface UCX II on my Mac. What else should I do?

Yes, it is listed in USB Device Tree.

You could try the kernel extension driver v3.28 once.

I did try that before and it worked.. Reinstalling v3.28 made everything work again.

But this is more like a last resort. I still wanted to figure out how to make the newer driver work.

The new driver requires UCX firmware to be 49. I used the Flash Tool, it shows my current USB version is 43, and the new version is also 43, so even after Flash All nothing is updated. I have tried the Flash Tool on a mac that runs the Kernel Extension driver and a Windows computer. Same result. How do I update USB version to 49?