Update: I went through the whole support rigamarole with Norton, and determined that Norton wasn't doing anything to the newly-downloaded versions of the trainers. The one useful thing the Norton tech had to offer was that the file might be corrupted.
In view of your successful test, PWizard, I downloaded the XCOM 2 trainer to my laptop, where it ran as normal. When I transferred *that* copy of the trainer to my main gaming rig (where all of the trouble is) it ran as normal.
The obvious conclusion is that there is something wrong with the download process specifically on my gaming rig. Do you have any thoughts about what that might be? I have a viable solution via my laptop, but it's a pain. Do you think my copy of WinRAR might have gone screwy?
If you have any kind of browser addons or extensions, especially any security related, even by Norton, then disable them as they can cause the downloads to be corrupted.