Disagree. McCarthy had widespread support outside the eastern media until it became clear that he didn't have the evidence.
Remember, the reason McCarthy was listened to in the first place was due to the Hiss perjury trial and conviction (which followed the Hiss/Chambers HUAC hearings). The possibility of Communist infiltration was clearly quite credible and was accepted by everyone but the liberal media. A careful investigation in 1951-52, led by someone like a then-Senator Nixon, would have rooted out the problem without undermining the movement.
Eisenhower got involved (subtly, in the background) when McCarthy started going after innocent Army people. Before then he frankly didn't want to touch the issue, because McCarthy had such widespread popularity, which goes against your third rail argument.
You're right that Ike didn't want to touch the issue. He had no idea how many cans of worms this was going to open up. He had already seen what Truman went through. And that is the sort of thing I was referring to as a "third rail." It's certainly no coincidence that McCarthy made the issue of communists in government his personal property -- nobody else wanted it. Even Nixon only took the Hiss matter so far. It was somebody else who sent Hiss away to Lewisburg.