I'm with you on the 4th band after the Beatles, Beach Boys & the Stones. I had a hard time deciding between the Bee Gees & KISS for the 4th spot. But I went with the Bee Gees.
Interesting that you didn't consider Zep for #4. Of course, I didn't consider the Beach Boys, so there we have it. I think (i.e., in my opinion), the BBs were very influential in that period between 62-67 but not sure they were ever as 'big' as the Stones, Zep, Beatles, et al. But, I can't argue that they were very popular and influential.
Of those I considered for #4, I left out AC/DC, Queen (were they ever as big in the States as they were outside?), BBs, Sabbath - a lot of good bands with longevity and huge followings.
On my Mt. Rushmore, only the Stones and perhaps #4, depending on who I slotted in there, would be among my favorite 4 bands. So, at least I stuck to the theme of the OP. Good thread.