Thank you, HappyScotsLass. Your suggestion didn't completely remedy ny situation, but it cleared up a number of things. My iMac is connected by Ethernet to my Airport, which is running both 2.4- and 5GHz networks, so I wondered if some combination of those things was messing me up.
So I followed your directions, including unplugging EVERYTHING from the iMac for several minutes, restarting the Airport and powering off and on the iPhone. After restarting, I attached the iMac to the 2.4GHz network (the same as the iPhone) and Voila!! It worked like a trained pig at the county fair! I synced several times, initiating from either end, all like clockwork.
Emboldened, I dared to switch the iMac to the 5GHz network and it still worked. Oh, Joy! Oh, Rapture!
So then I returned to the setup that I actually use: I just plugged the Ethernet cable into the iMac and the next sync attempt failed, and failed, and failed...
So this is me, turning off wi-fi on the iMac and officially giving up.
P.S. I guess USB syncing isn't so bad, but I just HATE it when an "up"grade (as in "up yours") takes a way a feature that I'd become accustomed to. If I wanted to fear every "up"grade I could go back to Windows.