Much to my disappointment, I found out why. FMServer 15 isn't event tested on Windows 2012 R2 Datacenter. To quote the very nice tech support person, "I'm sorry. That sucks." Yes. Yes, it does.
So I suppose trying it on Foundation Edition would be a waste of time, too? I'm currently running FM Server 14 on Windows 8.1 Pro - maybe I'll just leave it alone for now.
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UPDATE to my original post.
The whole server was sluggish, not just Filemaker. The IT manager pounded away at possible causes until she found that the Broadcom network cards have a (somewhat) known driver issue that creates a stranglehold on data flow. She fixed the drivers and everything, INCLUDING FILEMAKER SERVER, is now superfast.
In the meantime, I'd eliminated deprecated themes and cleaned up a number of calculation fields to improve speed, which undoubtedly helped, but getting the right drivers made all the difference.