Well, naturally, I didn’t send in the phone for hardware analysis, so I suppose I should say that they (and I) determined that it was probably the antenna or radio. We started with profile and PRL updates, then a backup and a “reset” of some sort, all the while I watched the Phone Status page. I did get a Republic Network connection very briefly once. They checked the IMEI info against something, which, I guess, was OK. I removed the sim card, photo’d it, and emailed the image, and they verified that the sim card was as it should be.
Then we did a this-time-we-really-mean-it reset (##786). Actually, I did a whole bunch of ##786 resets and closely watched and photo’d the network status and signal strength. A couple minutes after each ##786 the phone automatically gets a new cell service configuration – “Cell configuration required” – and reboots itself. But one time after the ##786 but before got a new cell configuration I captured a photo with signal strength -103 dBm 37 asu while my Moto G1 had about 82 dBm 2 asu. At that point the tech said the problem was either an issue with the cellular subscription or the antenna. So he assigned a new cellular subscription for the phone (somewhere in the Sprint/Republic system) and then I updated the Profile (after first cycling power once again). The new cell subscription didn’t help, so we concluded it was the antenna. (I say it could also be in the radio, but maybe that’s much rarer than the antenna.)
I wrote that I’d probably buy a Moto G4 Play, preferably through RW, but since RW doesn’t carry it I’d buy it elsewhere. The tech responded with a couple good ideas for getting the Moto X1 repaired, and he said that I’d need a new sim card for the new GSM network. Today I bought the Goto G4 Play (elsewhere) and the sim card.
So I guess the answer to “how did they determine it was the phones antenna?” is that we didn’t absolutely determine it, and that the Moto G1 helped build confidence – at least it boosted my confidence.