Thanks for all of the feedback -- it's useful.
A couple clarifications: we do have redundant systems, on multiple physical machines with redundant power and network connections. If a VM (or even an entire hypervisor) dies, we're generally OK. Unfortunately, some things are very hard to make HA. If a primary database server needs to be rebooted, generally downtime is required. We do have a pretty good monitoring setup, and we also have support staff that work all shifts, so there's always someone around who could be tasked with 'call me if this breaks'. We also have a senior engineer on call at all times. Lately it's been pretty quiet because stuff mostly just works.
Basically, up to this point we haven't automated anything that will / could be done during a maintenance window that causes downtime on a public facing service, and I can understand the reasoning behind that, but we also have lab and QA environments that are getting closer to what we have in production. They're not quite there yet, but when we get there, automating something like this could be an interesting way to go. We're already starting to use Ansible, but that's not completely baked in yet and will probably take several months.
My interest in doing this is partly that sleep is nice, but really, if I'm doing maintenance at 5:30 AM for a window that has to be announced weeks ahead of time, I'm a single point of failure, and I don't really like that. Plus, considering the number of systems we have, the benefits of automating this particular scenario are significant. Proper testing is required, but proper testing (which can also be automated) can be used to ensure that our lab environments do actually match production (unit tests can be baked in). Initially it will take more time, but in the long run anything that can eliminate human error is good, particularly at odd hours.
Somewhat related, about a year ago, my cat redeployed a service. I was up for an early morning window and pre staged a few commands chained with &&'s, went downstairs to make coffee and came back to find that the work had been done. Too early. My cat was hanging out on the desk. The first key he hit was "enter" followed by a bunch of garbage, so my commands were faithfully executed. It didn't cause any serious trouble, but it could have under different circumstances. Anyway, thanks for the useful feedback :)