Somebody wrote an expose about the Danny Thomas set, and the troubles there - I think it was Jean Hagen, who played Danny's wife first - before Marjorie Lord. Whoever it was indicated that Hamer was an obnoxiously, spoiled, entitled brat - and very difficult to work with. I suspect he had greedy handlers (his parents perhaps) who were responsible for his behavior. It's not surprising his career cratered after MRFD.
A few years ago, Sherry Jackson (who played the daughter on the show--until leaving after 5+ seasons) basically confirmed that Rusty Hamer was allowed to get away with murder on the set because he was a cute kid, etc. When he was no longer that cute kid, he got a pretty heavy dose of reality. She also told the story about how Hagen hated Thomas (one reason why she left after the first three seasons) and on one occasion, Jackson grumbled to her about something that Thomas did and she responded, "You don't have it bad, I have to kiss him!"