It's not about whether a browser renders the page. It's nothing to do with obsolescence. The frequent updates aren't for functionality, they are for security.
It's similar to the Windows security updates which are at least monthly, and more frequent for critical issues. Many of these are internet explorer updates. Don't confuse the little updates with the big version upgrades (e.g. IE8 to IE9). Other browsers too have frequent security updates.
So, as a tech you should indeed be trackng every tiny update as rapidly as possible, to avoid exposure to malicious attacks, which are often barely days behind the fix.
(Not an Adobe reply. You might consider posting Flash questions in the forums for Flash, rather than Acrobat.)
By the way... if you aren't equipped to deal with these frequent updates, you should be considering allowing Flash to auto-update. This new-ish feature I found most welcome.
One more note. As an manager of multiple systems perhaps you are already using enterprise deployment tools to manage updates etc. If so, you may be able to integrate Flash into these tools so you can maintain the same level of preparedness against threats. http://www.adobe.com/devnet/flashplayer/enterprise_deployment.html
On the other hand you might take the view that the greatest threat to the security of computers in a school comes from those who sit at them... depends whether these are student-side or back office.
What is SO critically important about having the bleeding edge version of Flash on the computer?
Help yourself > http://www.adobe.com/support/security/bulletins/apsb12-19.html
FlashreaderisFrustrating is right. I have many machines where I have to update them all the time. Automatic updates are not a solution since any update to a machine is a risk and can change its behaviour. (All of a sudden a flash that worked now fails. How can you back out of an automatic update and don't even know if it happened. Adobe needs to stop and fix the software correctly and securly and provide updates (i.e. quarterly).
I assume that you also do not apply the monthly Microsoft Security Updates?