Your suggestion that I connect the analog to ground was the suggestion I needed. It confirmed that the board and software worked, and also forced me to re-connect things.
Here is the full story, if it may help others who go down this path...
Yesterday, I had I confirmed that the resistance of the pot is what it should be across the end terminals. Also, confirmed that the wiper is behaving as expected for this linear-taper pot. So, mid-way is mid-way in terms of resistance.
When the circuit was live, the voltage at the wiper is also as expected.
However, I had not tried connecting the analog pin to ground. But, my test probes and alligator clips were in the way, so I decided to just disconnect them and try to start with everything as simple as possible. I had actually already done a re-work of the connections yesterday probably three times.
The test showed that it did, indeed show the expected 0 value when the analog pin was connected to ground. So, my Mega was working! I re-connected everything to test, and was surprised that the values now only ranged from 0 to 454! Re-checking the connections, I saw that one of jumpers that I thought was to 5v, was actually on the 3.3v pin.
It can be difficult to see which pin corresponds to which label on my Mega. In the attempt yesterday, I am taking a wild guess that I may have connected the pot to 5v and 3.3v. So, "low" would have been 3.3v.
Thank you for taking the time to try to remotely trouble-shoot my situation.