I tried a different cable which did not help. I just got a 4.0 and a 4.1. I have a lot of experience with the 3.2. I'll try on another computer running BigSur
So far the only reports of trouble are with Apple's beta versions. It definitely works with the officially released Big Sur.
As you can see on that thread, I updated my Macbook Air a couple days ago and personally tested it with Big Sur version 11.0.1. Works fine.
I know you said you tried another cable, but I'd like to repeat that by far the most common problem is charge-only cables. More than once we had someone try as many as 5 different cables from their drawer / box of leftover cables, and all of them turned out to be leftover charge-only cables. Seriously, get a known-good cable.