On the flip side, spending six weeks fixing an issue on a single server running a non-critical, non-time-sensitive service which occurs once or twice a year and is 100% worked around by a reboot probably isn't an efficient use of your time.
In the long-term, it is. If you let issues like that continue to exist, then you'll get stuck with an unnecessary proliferation of servers, with each running just one service, so rebooting one doesn't take the others down.
Not to mention that you'll find that you get stuck maintaining multiple, overlapping services, because the first one wasn't reliable enough for the tasks some department decided to bring-in, later.
And also, I don't think I've ever seen a service that was non-critical and non-time-sensitive. Whatever it is, people won't even try to use it until the very last minute, when they need it to work immediately. It could be a damn web page that just hosts the phone extension list, and because HR needs to call someone about something simple, at 5pm on Friday, that server is now delaying everyone's paychecks. EVERYTHING ends up being varying degrees of critical.