By the 60's Rock was inherently a different beast. It was called Rock and Roll rather than Rockabilly. Albums weren't called race records because they were involved all-black bands. The entire reason Elvis became anything was because his record label looked and him and said, "We can use this guy to sell the records America is uneasy about because it's all being done by black people to sell it to them worry-free because it's a white guy who goes to church every Sunday."

It'd never entirely get away from it's image as promoting sex and drugs (rock and roll was slang for sex after all) but for the most part Nixon saying he liked Elvis would be, politically, a ground ball. Not terribly controversial. I mean, he's the best selling solo artist in the history of recorded music.

Furthermore you need to appreciate the overall era, and what the two represented.

First- Elvis wasn't someone who took kindly to the growing hippie culture and the wider drug culture. This was likely more a business deal- he was the old vanguard dealing with the British invasion poaching his business at the time. His wider abuse of drugs wouldn't come out till after Nixon ducked out. Elvis's marriage wasn't on the rocks- at least not publicly- yet.

Second- Nixon needed to maintain his political appeal. Part of what got him elected was the promise to pull out of Vietnam. Contrary to what people realized then- and today for that matter- you don't simply pack your bags and leave. At the same time he was a bit....eccentric? Go to his wikipedia page and read up on the "personality and public relations" tab. It describes him rather well- needless to say he wasn't exactly a charismatic master of PR.

For Nixon, saying he liked Elvis was about image. Elvis, at least until....72, 73, 74 was about as beige as you could get for a rock artist. He was no Johnny Cash. His image was supposed to be that of the good Jesus boy. That image wouldn't exactly change until the skeletons came out of his closet- but by that time Nixon was out of office.