The same support person got back to me. He apologized not having gotten back to me sooner. Also he seemed to suggest there might be complementary support if I called Apple Care. It seems clear that the express lane does not really offer technical support for the iPhone.
Here's the bad news. Seaottr's suggestion worked for me. Thanks so much Seaottr. I'm now going to have a hard time continuing an investigation which might help others.
Since I arrogantly said we'd like to understand why rather than just get a solution, I feel I now owe a guess at why the fix worked. Please ignore the following if you don't care :-) Also, I'm probably wrong.
Since I had three phones with the problem I had a bit of an opportunity to try some variations of Seaottr's solution. First time I followed Seaottr's suggestions to the letter. It worked. The second time I didn't bother with the time zone. It worked. The third time all I did was set the year to 2020. It worked.
In the past when I'd followed the advice of date changes I'd never been instructed to go more than a couple of years into the future - so I never did.
It seems like my problem was an SSL one (or something like SSL). Changing the time way into the future invalidates Apple's server certificate. That certificate is used to prove to the phone that it is connecting to the real Apple store. Any certificate like that has an expiration date. It's up to a device to download a new one when the old one expires. Or it's up to Apple to provide a new one; perhaps with a OS update.
So setting my time way into the future made the phone think the certificate had expired. It probably then deleted the expired certificate. Then I set the time back to now. The next time an attempt was made to access the store the current certificate was redownloaded. Perhaps that certificate has an expiration date 3 years in the future so in the past just setting my date forward by a couple of years didn't trigger the expiration and discarding of the certificate.
But how did the old certificate that had to be dumped get bad? Since it happened to 3 of my phones there had to be some relationship between those phones. Since my three phones use the same AppleId I'd have to guess that was it. There's really nothing else in common. There might be some coupling of the AppleId together with the certificate which has to be in a valid state. So if something was wrong with my AppleId then it could have caused a problem on all my phones. It's kind of typical to do things like "signing" an AppleId with a certificate. That's the kind of coupling to which I'm referring.
So what makes me think I had a bad AppleId? Well, I hadn't used an email address for my very old Apple Id. Futher the password I had used was invalid by Apple's standard.
Apple's password standards must have changed recently. The old password I had wasn't more than 6 months old. On a lark I went to http://appleid.apple.com to change my id to an email address. However the existing password I had would not be accepted; it was not sufficiently complex. So I made it more complex. (Interesting that I had read posts elsewhere that changing the password on an AppleId has fixed our problem for some.)
So perhaps, Apple was rolling out a new certificate over a period of a week to all their devices. As the new certificate got to each of my phones the bad AppleId triggered a problem. Or they were rolling out some code relating to the tightened password requirements on AppleIds which triggered the problem (that seems more likely).
So, fixing my AppleId and THEN forcing a redownload of the Apple server certificate solved the problem.
I have a 25% confidence level that 80% of this is correct.
