After a few minutes of experimentation, I discovered that Synergy was to blame. (For those unfamiliar with Synergy, it's a software-based KVM (non-video) system to share keyboard/mouse/clipboard across multiple platforms. While it was originally open-source, it has become a paid-download product - one of the VERY few such packages I use.) As it happens, Synergy's default behavior is to synchronize screen savers across all participating systems; since my Synergy server (running Linux) did NOT have a screensaver enabled, the Synergy client on my Windows 7 system silently disabled its screensaver as soon as it made the initial Synergy connection.
Well, now that I know this, how do I fix it? Well, there's some tweaking to be done on both sides.
On the Synergy server, I added the following section to /etc/synergy.conf:
This disabled the synchronization of screensavers across my Synergy systems; however, I immediately found that I couldn't send a control-alt-delete signal to Windows 7 via Synergy. Well, it turns out that, by default, Windows 7 doesn't allow the Secure Attention Sequence (SAS - the fancy name for control-alt-delete) to be generated by anything other than the system keyboard...but there's a fix for that. Head over to the Start menu and run gpedit.msc (the Local Group Policy editor). Navigate through Computer Configuration, Administrative Templates, Windows Components, and Windows Logon Options. At this point, double click on "Disable or enable software Secure Attention Sequence" and configure the policy like this:section: options
screenSaverSync = false
Once this policy was enabled and applied, everything worked as expected; my Windows screensaver kicked in when it should AND was password-locked, and I could send Control-Alt-Delete via Synergy with Control-Alt-Pause/Break. A quick security scan confirmed that my employer's requirements were met, and all is now well.
I'm not running Windows 8 or Windows 10 in my environment; if you ran into similar problems with Synergy, let us know in the comments.