I saw something on ESPN that debated where that whole "America's Team" garbage came up in the first place.
They mentioned whether the Packers should be America's Team now, since the Cowboys have been irrelevant since the 1990s. The argument against was that most people like the Packers, and being America's Team requires a lot of hate and being the most nationally talked about team.
I always resented why the Cowboys could declare themselves "America's Team" -- I never voted for them. But unlike the Pack, they still generate a lot of hate. But like Juve, years of underperformance can turn hate and respect into sympathy.
And the Cowboys may not have much sympathy, but they're hardly relevant to playoff discussions for a generation now.