So you can now no longer listen to Radio at all unless you join Apple Music?

The old Internet Radio stations are still available, independently of Apple Radio. I can't answer for older versions of iTunes on a Mac, but in the current one you can access these by clicking on the three dots icon at left of the toolbar and choosing 'Internet Radio'.

If you click on the Music icon in the toolbar and then click Radio you get to the new stations; non-subscribers can access Beats 1 radio but the Featured Stations shown are only available when subscribing.

Actually I think it's the other way around. The Featured Stations and Recently Played work on my devices. But I believe you have to sign up for Beats 1.

No, I just tried it to make sure. I don't have Apple Music, but Beats 1 works OK. The Featured Stations invite me to join Apple Music.

I know why. This MBP is running 10.7.5 so Beats 1 isn't available. No Apple Music here either but the Featured Stations do play and also on my iPad and iPod running iOS 8.4. That seems kind of strange.

Are you sure you have the Featured Stations and not just the Internet Radio stations? The Apple Radio page looks like this:

Of course I what version of iTunes you have - this is 12.2, which will run on 10.7.5 (though Apple Music won't, as you say).

Yes that's what I see. There's also Recently Played which still shows some of the Artist based radio stations I made quite a while back. That Sound System station is one that I checked earlier. When I click on Beats 1 Listen Now, there's a pop up that says I need 10.9.3 or later to play. I have iTunes 12.2. I think 10.7.5 is just barely "new" enough for iTunes 12.

Maybe for newer operating systems there's no option to hear those stations without paying and getting it commercial free. I have occasional commercials here.