"Pride" being understood as something everyone has: in their country, in their family, in their diligence or hard work or history; an antidote to shame.
You were right until the last part. It is an argument with wide appeal, but that appeal is to people's lowest instincts.
Let's consider a similar argument that is made: people argue that sexuality is not a choice, people are "born this way", that it's an innate characteristic. The power of this statement in context is that it takes advantage of people's bigotry.
Gay Person: I was born this way.
Homophobe: I think you could choose not to be.
Gay Person: Could you choose to not be straight?
Homophobe: Of course not, I could never be gay.
Gay Person: So it's not a choice.
It's not a winning argument because it reasons to a conclusion that others will accept based on their beliefs, but rather because it gets almost anyone who would challenge it to refuse to answer honestly based on their own prejudices and insecurities.
"We're proud of who we are" works in a similar way, it's not an argument that reasons to a conclusion, but rather one that sticks by consequence of people refusing to set aside their own ego. It's not elevating everyone, it's just acknowledging that we all suck together. Which is certainly true, but not what you want to have a celebration about. Yes, the world is full of people proud of their nationality, gender, race, etc. That's all bad. It's not an antidote to shame, it's a source of hatred. Parents feel pride when their children win an event, and chasing after that feeling leads to horrible parents that spew vile things at their children's opponents or the other parents or the judges ore referees involved. A world where everyone is proud of themselves all the time may or may not have less shame involved, but it's also a world where everyone hates each other all the time.