It turns out you can move a "Recorded Meeting" to the "Content Area" of your Connect account, so that Connect will treat the recorded version as "Content". Once you move it, you will now have access to the report section for the "content" (including the count of users who accessed the recorded version).
(Thanks to "Alain" on the Connect forums for sharing that answer when I asked about the challenge.)
Once you move it, you may wonder where to find it in the Connect Admin interface. You generally work in the "meetings" area, but you'll see a link in the top navbar next to that saying "content". That's where the recorded meeting is moved. Any URL for the recording remains the same.
What's more the count reflects all views since you recorded it, not just since you moved it.
One gotcha to watch out for: be sure to check the permissions on the recording, as the move could change the permissions. (Update 11/07: there was a time when I observed that the permissions would change, but recently I observe that I no longer need to do that.) Check "set permissions" while looking at the list of meetings at Content>My Content.
For more on the notion of user group managers using Connect, see the links below.