I would venture a guess that most of those who feel the Eagles are a bigger rival are the younger fans who didn't grow up hating the George Allen coached Commanders. He personally built that rivalry into the most heated in all of pro sports.
Allen HATED the Cowboys and was obsessed with everything we did. He built his teams specifically to play us and sacrificed high draft picks every year to bring in guys who could help him beat the Cowboys.
The sad thing was that the Commanders only went to the SB one time under Allen and lost (1972) that was also the only time they beat us in the playoffs (whooped us bad that day though, 26-3).
I believe that Allen sacrificed team success in order to beat the Cowboys. They couldn't beat the 49ERs, Vikings, or Rams in the playoffs under Allen because his team was built for one purpose only, to beat the Cowboys.
We split the regular season matchups every year he was the HC but his greatest victory was his single playoff win over us in 1972. He used to crow about that game all the time too.