Recording in breakout rooms is only possible when someone is in that room directly and recording locally. If the host is in another room or the main session, the host would need to give one of the breakout room participants the ability to record locally to capture that room's audio and video.
I was just looking into this for my choir as it returns to Zoom rehearsals with sectional breakouts and conclude the following:
Recording Breakout Rooms
Zoom does not allow a single account to record more than one stream at a time to the cloud storage although it does allow the same account to be logged in multiple times and each can record to local storage.
I came across this procedure that can allow a single computer to record multiple breakouts locally but it may be challenging for the computer, its internet connection and the operator. If the system crashes, all is lost!
"If you need to record all of your breakout rooms, it is recommended to give individuals the ability to record and designate someone as the videographer for their room." Recording a Zoom breakout room, University of Iowa Office of Teaching, Learning & Technology. Depends on each section videographer but one crash or forget loses only one breakout room recording. More logistics involved in gathering the recordings than for (1); compiling and posting effort probably the same. An alternative to putting breakout rooms up on the website would be for each section videographer to make their recording available to their section members by whatever means.
I suppose we could attempt both approaches with (1) as primary and (2) as backup