OK, after a bit of digging, I think I’ve figured out what is going on.
The standard MQTT implementation only requires MQTT brokers to accept client IDs of up to 23 characters. See http://docs.oasis-open.org/mqtt/mqtt/v3.1.1/csprd02/mqtt-v3.1.1-csprd02.html#_Toc385349767 for details.
It turns out that the MQTT client program I was using (MQTT-Spy, for anyone interested) automatically truncates supplied Client IDs beyond 23 characters. I guess this isn’t strictly required, but it is based on the quasi-standard indicated above (i.e. that there are brokers out there that would not accept a Client ID longer than 23 characters).
As the Cayenne Client ID is longer than 23 characters, the Cayenne MQTT broker was apparently not “seeing” my MQTT client. When I tried with another MQTT client software app that does not truncate IDs, everything worked, I got my Cayenne dashboard, and was able to post “fake data” to it.