I agree with you that the rockabilly edge in rock music faded in the 60’s, but it never disappeared. Jerry Lee Lewis and Carl Perkins continued to play their music all through the late 50s and 60s, except when they were in the hospital. The Beatles covered some of Carl’s music and George was a big fan of his.
You can find compilations of all the rockabilly from the 50s and 60s you want.
Then there was the Allman Brothers and the rise of southern rock, which had definite country influences. The Byrds did country music. Linda Ronstadt covered Hank Williams. Creedence had country influences big time.
I think what happened was the British bands weren’t able to do country as well as they adapted blues and R&B, and British bands were a big part of 60s rock.
The Band were unique. No doubt about that, but I don’t think they were the only rock band with country roots showing.