OK so that sounds much like what limited iTunes all these years, you had to manually toggle the damn Audio Midi Setup setting in Utilities to get the sample rate to change, which is ridiculous.It is connected via USB, and I am indeed running 11.4
One way around that was to purchase the BitPerfect iTunes plugin, I'd be curious if that also works with Apple Music. I'm still running macOS Catalina 10.15.7 or I'd test it, as I do have BitPerfect.
Odd that Apple would make no mention of any of this, and what they did say was you need an external DAC, which would imply thats all that you need.
I just launched BitPerfect and it does work with the version of the Music (iTunes replacement) app that I have, however that is for local file playback, I'm not an Apple Music cloud service subscriber.
If you are willing to risk the $9.99 on BitPerfect, that may just work, and even if it doesn't, it should allow for automatic sample rate switching for local file playback unless incompatible with Big Sur.