There are times where staffing is reallocated elsewhere in the mornings, from what my CM friend said it is often due to callouts from other CMs. That said of course there is a reason Disney wouldn't want staffing issues to be front and center to the guest's notification so I wouldn't say "there'd be no reason to keep that information secret" there are a lot of reasons why they wouldn't want to disclose a lot of things and is why they don't disclose a lot of things. We know a lot more stuff than is publicly announced here on the Boards for example just by having a few insiders (like Len Testa for example) than will ever be publicly acknowledged by Disney.
As far as less popular rides they could keep closed Disney (or any theme park company) isn't necessarily going to use popularity of an attraction for what they opt to do. You could, and this is purely just to give another perspective, see it from the standpoint of low wait attractions that a guest can get more attractions ridden in less time thereby being more satisfied with their park time. That is a strategy many employ in the mornings whether it is Early Entry or just park opening, you could go for the attraction that may garner higher waits later on but by doing so sacrifice the ability to go on even lower wait time attractions and multiple of them in the same amount of time you waited for the one attraction. That shouldn't be construed as saying Disney cuts staff at 7DMT, etc just speaking to the "less popular rides" part of your statement
