I opened a ticket with Apple using the Feedback Assistant on Sept. 15th 2022.
I updated it today. No responses to it, ever.
According to the notes, I'm the only one that reported such an issue.
The evidence I had then, and still have, strongly suggests this is the result of a change Apple made between IOS 15.x and IOS 16.x.
I have an old iPhone running iOS 15.x and I have a MacBook Pro that does not support the latest macOS (and so runs an older version of Apple Music).
On each of these devices, Apple Music will play all RTE Radio stations.
On the iPhone (iOS 15), TuneIn Radio (currently 24.8.1) will play all RTE Radio stations.
The same was true for an iPad before it was upgraded to IOS16.x.
To the contrary, Apple Music and TuneIn Radio (24.8.1) on IOS 16.x will not play any RTE Radio station.
There was a version of TuneIn that did work on IOS 16 for a short time but that was upgraded soon after IOS16 was released after which TuneIn no longer worked.
As far as I can tell, any IOS16.x device (my Homepod Mini is impacted) will not play RTE Radio but IOS 15.x devices will continue to do so. No idea why only RTE is impacted but I'd like Apple to explain.