The Cowboy fan-base was built before the Cards came to AZ...AZ was in the Dallas regional broadcasting area...Once the Cards came, fans from New Mexico would come to AZ for the games, because PHX was closer than Dallas. The large fan-base hasn't gone away, it's just the Cards winning and the new stadium has made it more difficult for them to overwhelm at the local games.
This ^
I was never a Cowboy fan per se, but my oldest son became one because that's who we saw play every season. Ironically, we also saw a lot of the Raider games on NBC back in the day and another son became a Raider fan.
I, on the other hand, was a fan of the St. Louis Cardinals. I don't know why, maybe it was Jim Hart and the Air Coryell days, maybe it was because Dad was a St. Louis Cardinals baseball fan. At any rate, I have loved the football Cards for over 45 years.
While I don't have the "hate" for the Cowboys (or any other team for that matter), I certainly never rooted for them. Other than the Cards, even before they came to the Valley, I was a fan more of players than teams I think.
So, to many the Cowboys were the local team here in the Valley for many years before the Cardinals showed up. For some, the loyalty runs deep and remains to this day.
Lucky for me, both of my boys became Cardinal fans once they came to the Valley. We all root for the home team, even when they were absolute suckage in prior years
Go Cards!