> The US Congress does, really, have line-item authority over regular US laws.

I don't think legality was the issue. The issue is that there are dozens of countries that need to vote on the same exact text for it to have an effect.

Surely the US Congress could modify it but then the other players will at best ratify a different version and at worse change items here and there themselves.

In both cases, what you get is not an international trade agrement. You may as well vote "no" or not even try to negotiate a deal.