Why is the Read-Only Routing not functioning in this Availability Group?
The human element was the problem, as usual. Turns out I had accidentally used the primary replica's server name in the Read-Only Routing URLs for all three replicas. There being only one character difference between each name made it difficult to spot, even though I'd looked at them dozens of times.

Thanks to @SeanGallardy for prompting me to pull the URLs, which caused me to notice the problem.

Thanks to @clifton_h for also making a good-faith effort to help.  Y'all good people!