I just finished the soldering and am about to reassemble my TV. Turns out, the problem was that the tip on the iron was shot. I bought a new $5 tip, and it now works exactly as it's supposed to. I replaced the two amplifier ICs (X and Y for each of the R, G, and B guns -- about 40 solder points in all) plus three resistors that were clearly charred to a crisp. Those resistors were burned sufficiently that the only way we know what to replace them with was to look at the Hitachi service manual, which I was thankfully able to find online. I love the Internet, I really do.

It's amazing how, once I had it all figured out, that I could blast through all those solder points in zero time flat. Now, fingers crossed as I head back to the TV...