So I came across something weird that's affected only my 10.12.4 clients (none of my 10.11.6 clients seem to be affected by this). Even though I have only one Safari profile enabled, which is set-once and doesn't manage the homepage, my 10.12.4 clients are unable to change the homepage in Safari manually. Whatever the homepage was is stuck like that. If you enter a new homepage in the Safari preferences, it will just not take and revert back to the old homepage once you hit Enter or click out of the address entry field.
The only workaround I've found for this is to delete all profiles (again, even though I don't have any profiles managing the Safari homepage):
Are you sure you want to delete all configuration profiles? [y/n]:y
Also, after testing on a couple of other clients, there do seem to be situations in which the Safari profile was never set at all, and you still can't modify the homepage, even after deleting any other profiles and rebooting, and it's not account-specific either (freshly created account experiences it, too). It's a real head-scratcher.