Strange and sorry upfront, if I should understand your problem not exactly...
Your english is fine btw, surely better compared to my Japanese :-)
For one thing, I've never had the problem that anything from YouTube was too quiet.
Could you kindly give some example URLs please, which show your problem, that it is too silent.
Unfortunately you didn't tell, which operating system you are using. I can only comment to a Windows system.
And as I don't know your setup exactly, just a few more "guesses" from my side....
Under Windows audio from a browser is being routed through the Windows sound system including Windows Mixer. I would double-check again, if you have turned down audio somewhere in the Windows Mixer or maybe there is something additional in the signal path of Windows audio.
If so, the output level of the software playback channel in the middle row of TotalMix would also be relatively low, or the video wouldn't have a higher volume. One of both.
If you would send an example URL, then we can cross check for you, how it sounds here and maybe this makes things also clearer. At the moment I have some doubts why the Babyface Pro should have an issue. Rather than that I would simply say, if Youtube Audio is too low, turn up the volume ...
BTW, to complete the informations ...
What OS please.
Where is it too silent, through monitors or phones ?
Which monitors and phones please (brand, modell) ?
What are the settings of the HW output channel ?
A screenshot of your TM FX settings would be fantastic
- showing the "submix" / HW output where this occurs
- showing the HW Outputs settings in TM FX
- showing the signal in the SW Playback Channel for this particular submix
BR Ramses - UFX III, 12Mic, XTC, ADI-2 Pro FS R BE, RayDAT, X10SRi-F, E5-1680v4, Win10Pro22H2, Cub14