[cdellin]

From the section you referenced:

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

I think that your claim that the only role of the Senate is that it "must be notified in the event the President has made a sole-executive agreement" is a misrepresentation. In fact, the Senate must vote to approve every potential treaty before it becomes valid.