Your previous post made a light go on after I posted. The NAS is set up to use OpenVPN, and the "normal" (non-VPN) network connection was set to a static IP. I've now set it to obtain an IP address automatically, though somehow that doesn't seem to be enough. Maybe a reboot is required. I have to switch between the DIR-655 and the R7000 for every try, so that's taking a while.
Anyway, thanks - I keep forgetting about that OpenVPN setup and often end up diagnosing for nothing (though I'm not done yet in this particular case). I believe the reason it worked in the PC is that it has a different network adapter and it used its default settings to get its IP dynamically. That threw me off at first.