After running CHKDSK, the 60GB hard drive was down to 55GB total (50GB free), so presumably a bunch of bad sectors have been walled off. I ran Windows Update again, and this time all of the updates loaded and it will (I hope) stop bugging me about them. I reinstalled the video drivers (VTI Mobility Radeon 9000) from the resource CD, and they installed without complaint. After rebooting again, I tried the machine on the TV. Nothing.
I confirmed that the S-Video will connect on my own laptop. A little more investigation showed that the drivers on my personal machine are newer than those on the replacement. So it was back to the web to find the most recent drivers, which I installed on the replacement and rebooted again. Still nothing.
I tried everything I could think of doing to get the configuration of the replacement as close to that of my personal machine as I could, short of swapping the hard drives. (I'm tempted to do this, too, after what eventually happened.) Then I go the idea of trying to connect the machine to the other television, the one in my bedroom. The connection process there is slightly different, as it runs S-Video to a box that converts it to a signal that will go through the TV cable, rather than converting it to RCA composite video.
Eureka! The driver said "there's a connection available" and when I activated it, the bedroom TV recognized the computer and drove video to it. This seems to confirm that (a) the S-Video port on the replacement works and (b) the video drivers are okay.
So it was back to the living room for another experiment. Because I have so many different kinds of video going on, I have a switchbox so I can select between the computer and the DVD player. The cables from those machines run to the switchbox, and an RCA video cable runs from the switchbox to the RCA port on the television. Instead of connecting the replacement PC to the switchbox, I ran its RCA cable directly to the port on the television. Success! Again, the computer said "connection available" and could drive video to it.
When I returned the cables to the switchbox configuration, I confirmed that my personal machine could still get signal, then tried the replacement. No joy: it denied that a connection existed.
While this process is highly annoying, and took me several hours to troubleshoot, at least I'm narrowing down the field of problems. Something about that specific computer's S-Video connection is not seeing things in one particular cable configuration, even though the other computer sees the connection without a problem. I even have a workaround if needed. However, I don't want to have to root around the back of the television very often. The RCA ports are on the back of the machine, and difficult to access -- not something I want to do regularly. I may try some more experiments tonight.