Something seems to be screwed up somehow.

Please include information about OS and RME driver / TM FX version.
And Information about your setup: Monitors and Phones connected to which audio output ?!

What I personally do not understand and what I couldn't reproduce here
(UFX+ here, which is a different device and has other ASIO driver)
1. that we see in this video audio on two SW Playback Channel: Ph3/4 AND AS1/2.
2. that audio signal is missing on Ph1/2, but still continuing to play on AS 1/2.

SW Playback Channel means "audio coming from PC" (either the OS or an Application).
Therefore Information / Screenshots about WDM Settings in RME driver settings (TAB "WDM Devices" and TAB "Speaker") and Windows Audio settings would have been interesting before you do further changes.

My proposal to you is to:
1. save your current TM FX settings, so that you can restore them easily at any time
2. reset TM FX mixer settings to bring TM FX into a defined state
3. perform a very basic setup and check, whether the issue is still there or not and which can be repeated in a few steps by RME if any error should still persist

to 1) save TM FX settings
1.1 TM FX -> File -> Snapshot save as
1.2 TM FX -> File -> Workspace save as
      Workspace file contains all settings, routing and window size/position, fader width, ..
1.3 TM FX -> Workspace Quick Select -> Save Current Workspace As
      The latter has the advantage that you can restore TM FX with one keystroke "ALT-1"

to 2) Mixer Reset
2.1 TM FX -> Options -> Reset Mix -> Total Reset
This also populates the TM FX Control Room with default Main Out (AN1/2) and Phones settings.
2.2 TM FX -> Options -> Reset Mix -> Set Inputs Mono / Outputs Stereo (ADM)
If you use ASIO direct monitoring this is the preferred channel setting I read in the forum
2.3 TM FX -> Options -> Reset Mix -> Straight playback with all to Main Out
This enables Audio from all SW playback channels to Main Out (AN1/2)

3. easy to reproduceable steps
3.1 Save this to the next free Workspace Quick Select Slot, e.g. #2 Mixer Reset (ADM)
3.2 In RME driver settings (RME Icon in Windows Notification Area, Windows Desktop at bottom right)
3.2.1 Set WDM devices
3.2.1.1 TAB "WDM devices": Analog 1+2
3.2.1.2 TAB "Speaker": Analog 1+2 (this creates a speaker symbol in Windows sound settings)
3.3 Windows Sound Settings:
3.3.1 Click to "Speaker Symbol (RME ...)" make it to windows default sound device
3.3.2 Soundschema: "no sounds"
3.4 Rightclick TM FX Icon in Windows Notification, select "Exit TM FX"
      This is that the last settings file will be written definitively with these settings.
3.5 Restart PC
3.6 Login, Open TM FX and Firefox, playback Youtube Video

Expected results - assumed you have monitors connected to AN 1/2 -
stable audio output visible on these two channels
- TM FX - middle  row - SW Playback Channel AN 1/2
- TM FX - bottom row - HW Output AN 1/2

3.7 Quick validation of the routing to submix "Main Out" (AN 1/2, to Monitors)
3.7.1 Click to HW output Main (AN 1/2) in TM FX Control room, this shows the submix for Main Out (AN 1/2)
3.7.2 Validate, that the fader of SW playback channel AN 1/2 is up at 0dB for this submix