The the late 1950's and early 60's were generally weak years for popular music-- so weak that four merely good songs stand out clearly as "great" amid the rolling surf of mediocrities and manufactured product.
Early rock'n'roll like "Heart Break Hotel" and "Sh-boom" had been supplanted by look-alike drones and novelty tunes issued by the recording industry as after-thoughts to feed a market they thought was sure to disappear shortly. Serious musicians went into folk or classical. The fascists went for country. The teeny-boppers when for Fabian and the Italian studs like... Frankie Avalon and Tony whatever.
So a very small number of outstanding singles made it onto the play lists:
Runaway (Del Shannon)
I Fought the Law (Bobby Fuller Four)
Needles and Pins (Searchers)
Suspicion (Terry Stafford)
The last-named is really a rather blatant rip-off of the Presley version-- one of the few Presley songs I really like. If you search the internet long and hard, you might find a recording of the song including stops and retakes and comments by the musicians, including Presley.
And I can't remember what the fourth one was. I'll think of it eventually. [I did. It was "Needles and Pins". Not really, but it will do.] It was not "Sherry", which is a novelty tune. In the meantime, to properly appreciate the achievement of the songs listed above, you have to spend a few hours listening to nothing but all of the 2:30 long singles by the Chiffons and the Shirelles or even the Supremes. Don't believe the revisionists who will try to tell you that the Chiffons were actually brilliant: they were always mediocre. As for the Supremes... you can't hurry love.
They got on the radio because the airwaves were not controlled by listener preferences but by the backroom deal, as it pretty well is today.