From my experience you need less Linux sysadmins to begin with. Its easier to do remote admin. So the TCO numbers Microsoft claims are usually bullshit.
You have thought about that in terms of doing machine-by-machine maintenance. A large school district has a similar topology to a large enterprise corporation - thousands of systems spread out over dozens or hundreds of sites, with dozens or hundreds of different user-types grouped by function, having various seemingly-arbitrary blocking and auditability rules, and possible liability for certain types of breach, etc.
For maintaining a farm of identical servers, I agree with you completely. For maintaining Grandma's desktop remotely, I agree with you completely. But for maintaining an enterprise desktop environment, Microsoft simply has the best tools for the job. Linux AD-via-Samba quite simply doesn't even come close for the convenience of centralized GP maintenance, and has aothing anywhere near the convenience of drag-and-drop group-based software installation (though Linux does have non-stock application deployment packages available, like Puppet, that partially fill that last point). Linux has nothing even remotely like (W)SUS. And those two alone count as complete showstoppers when it comes to minimizing the number of people required to maintain a large network.
I love Linux, I use Linux, but Linux at the enterprise scale amounts to a non-starter.
Of course, the biggest irony here, school districts don't tend to use Windows, either - They loooove them some Apple products, which have all the same problems described above, plus the pricetag (not saying Apples still cost more, but they don't come free). So in that sense, yes, I can see how Linux would save school districts a hefty chunk of money; at some scale, however, you'll find that switching to MS would likely save money vs the overhead of sys/net ops and helpdesk staff.