[semimojo]

That is how the Founders wanted it. Equal branches of government having equal input into any agreements. The President does not have the power to negotiate on behalf of this nation by himself. Congress gets to add their input....especially at the end, before the deal is complete....and certainly not in a "take it or leave it" vote.

How would this work in practice? Would each of the 535 have a seat at the table? How could anyone actually determine the US's position?

There's a reason the executive is given the authority to negotiate for the US, subject to the input and approval of congress.

[Erik Latranyi]

Article 2, Section II US Constitution:

...He [POTUS]shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur;

Yes, Congress can amend it as much as they want.