Elvis has some rockabilly elements in some of his work for sure. However, I think that it is really pretty dumb that latter-day former 1950s rock 'n' rollers somehow converted Elvis to the country music category after he passed away. Elvis never was classic country in any respect....not even close. Still, lots of classic country radio stations regularly play many of his songs but I would contend that works like Hound Dog and Jailhouse Rock do not fit into the country realm in any fashion whatsoever. Of course, neither do the works of Taylor Swift or Darius Rucker and they both are huge Nashville country music stars. Go figure....I'm not even sure that I know what country music is anymore. If you are in favor with the CMA and the major Nashville record moguls, I guess you qualify. Personally, I'm much more attracted to the Austin sound as being representative of what real country music is than what Nashville churns out these days.
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I think that is due to the still existing party even racist radio bias. Back when Michael Jackson started taking off with his solo career, I remember reading an article in "Rolling Stone", german edition about this. Black musicians were automatically in the Rythm&Blues charts, white artists went into Country or Pop, and he was the major black artist who crossed over, and not only crossed over but made it to "King of Pop". But I digress.
I think country and other specific radio stations do not cover specifically country music, they cover music that will elevate their ratings, regarding the listening core. I worked in a small company here which made interviews over the phone with random people we wuld call and play them small snippets of playlists and they could decide which one they loved best. It was for the partaking radio stations to know their listener base, so they could best sell ad time to companies that viewed those specific viewers as their core buyers.
So taking Elvis in is not making him country. It's just he still guarantees a bigger listening - and buying the advertised products - listening crowd.