"So I have 2 x 1099 distribution forms from broker B for 2020, one for $6k"

The original topic is here: https://proconnect.intuit.com/community/tax-talk/discussion/excess-back-door-roth-contribution/00/12...

So, it turned out the May and Dec were both for 2020? And the Dec broker issued the Corrected 1099-R? And as I recall (instead of updating that topic where you have the background), you should have one of these (the $0) marked as Corrected.

"Do I need to enter both into his 2020 taxes as the corrected 1099 is zero?"

Nope. Nothing happened = $0. That corrected one is the failed conversion. You have one conversion to report, and perhaps some taxable earnings on the excess.