In answer to the questions...
Yesterday I spoke with the manufacturer. With their laborious assistance, I was able to ensure that the receiver is properly connected to the system. This in turn enabled me to install their driver. I've rebooted a few times with no change in results.
The batteries came with the remote, so I imagine they're OK, but I'll pull out my multimeter and double-check them. There is indeed something blocking line-of-sight to the computer (I'll deal with this later) but I went around to the back, pointed the remote a few inches away from the receiver, and...same result.
I guess what I was hoping for from the forum is some way to tell whether I have the appropriate (non-driver) software installed. That is, do I need to download something from Microsoft or elsewhere that "listens" to this remote? I'm (obviously) new to a lot of this, so I don't really know what I'm supposed to expect to see happen.