Fundamentally, threatening to walk away only works if people believe you'll actually do it. (It's plausible Trump has additional influence here solely based on the fact him doing crazy %$&@ is considered "plausible".)

What would be the practical effect if the U.S. had left? Having to negotiate fees separately with every country?

I don't doubt they could do it if necessary. Disruption would probably be minimal. But like I said, I don't think it was ever on the table.

I’m genuinely asking, is that what you have to do if you’re not part of this postal union or is there some other state you find yourself in as an outsider? Do packages stop getting routed via your country? I don’t know!