Another big issue was the time of the Monroe song - Monroe waltzed it and Elvis did it 4/4. At the time country was family stuff - waltz was docile; Elvis sold the 4/4 a bit more aggressively.
Years back, I remember hearing Buck Owens say that Chuck Berry’s “Memphis, Tennessee” should be regarded as one of the greatest country songs of all time- and he had a point.
Damn straight - and listen to Buck and tell me he doesn’t get rock songcraft. I have Dwight Yoakum’s (sp?) Dwight Sings Buck and freakin’ love it.
As noted above, yes, Monroe’s original version was slow waltz all the way through, but after Elvis’s version came out, Monroe re-recorded the song, starting out with a more uptempo waltz and then adding a driving 4/4 section at the end. It is this later version of the song that actually became more popular (and better known) with Bill Monroe fans.
Hope you don’t mind I actually spelled out why from a musical standpoint 
I’d have to add that Elvis was on the Louisiana Hayride for over a year. He changed that shows audience from older country fans to young rockabilly fans. The Hayride had one of it’s greatest years.
After Elvis left the Hayride died off. They had lost the old country fans during the Elvis days and there wasn’t much to bring in the young kids without Elvis.
I never considered Elvis country at all. He stayed much truer to Rock than Johnny Cash. Cash came from that same Sun Records rockabilly label. But Cash went totally country very quickly.
Maybe it’s got to do with who listened to Elvis then (pret’near everybody) vs who listens to him now (folks of a certain generation that trends conservative musically).