I tried a couple of them, and they both started downloading my entire backlog of email to my hard drive, which I didn't want.
I couldn't think of a reason why this would be necessary, but I haven't really kept up with how the technology has evolved in recent years. Is this behavior intrinsic to desktop clients?
Intrinsic, no. Common, yes. Many people who use desktop clients want a local copy of a substantial fraction of their email so that they can review or compose messages while off-line. Desktop clients also operate faster and can provide robust search services only if they have a cached copy of the messages on disk.
I can think of reasons why I might want a local copy, but they didn't apply in my case.
Do I have control over my data? I'm not sure I understand the question, but in this case the answer seems like a clear no, as my employer manages the email server.