Then I used Windows System Restore to roll the computer back to early Thursday before I installed 8.5. That got 8.5 off the system, but now 7.5 wouldn't initialize. At that point, I had no anti-virus/firewall software at all, so I gave VPN a try, and it worked. I shut that off and reinstalled 7.5 from backups. After it installed, configured, and loaded the latest patches and definition upgrades -- I went to have breakfast while it considered all that -- I launched VPN again, and it connected properly.
Despite the fact that AVG's tech support people say that if none of the steps they suggest work, the problem isn't their software, I should think that this experiment, with the only variable being 7.5/8.5 versions of their software, would be pretty persuasive that there is some significant negative interaction between their new program and my VPN software.