Problem solved, most embarassingly

Well, I have managed to fix my routing problem, and I'm embarassed to admit that it seems to have been something with the router. Nobody's quite sure what, since we had tried that route before, but a simple reboot of the household router and the problem went away. Since I have one of those router/nat/firewall combos, it could have been any one of those features. That doesn't explain why the problem persisted when I took down the firewall/NAT server, but it does seem to have cleared up nicely.

At least we know that if we see this problem again there'll be a fix for it. Even if that fix is as simple as "reboot the machine," something we've never had to do in the three years we've been using it.