From what I could find the adapter has to specifically have an integrated DAC(Digital to Analog Converter).
You can read a full discussion here that applies for the S21FE so it is safe to assume it will apply to the S21 Ultra.
I have a Note 10 and observed one day that I got prompted for a firmware update for the wired USB-C headphones that came included with the phone. This shows that they are not using an analog signal and there is some sort of integrated chip in the connector(I always thought that they were just simple analog headphones).
Picture of the headphones so you can see that they are the most basic of basic headphones. :)
This article was also helpful when searching why the issue would occur. It explains how analog audio works on USB-C when implemented.
The problem might be that the adapter that you have uses the analog implementation for audio but the phone only supports digital.
Also check the Media Output in the dropdown bar, as outputting audio should not require you to enter USSD codes(**#0808#* that you mentioned in your post.) I attached a photo of how it looks on my phone and highlighted it.
I don't have an adapter to test this but in my opinion this might be the cause.
I hope this is helpful and you manage to fix the issue.