I've listened to a fair number of the Mysterious Traveler radio shows and it seems to me he's always just the host and the narrator and doesn't intervene in the story.
Having him participate in the story, in the comics, seems like what was done earlier with the Shadow, where his role was expanded for the pulps and that in turn changed his role on radio.
DC seems to have done the same thing with the Phantom Stranger--although he never had a radio show. I find this a bit clumsy--like they weren't sure what to do with the Stranger when he appeared in the stories. Juat having him as a host would have been better.
I gather DC wanted to capitalize on the success of suspense radio hosts like the Mysterious Traveler, but I don't know why they didn't just license another host of a radio suspense show--such as the Whistler--rather than creating their own. They were licensing a lot of other radio shows at the time, so it was more than possible.