And yet, the Constitution also gives the Senate a check on the negotiating authority of the Chief Executive through its requirement to "advise and consent" to treaties with other nations, and the House has the authority to block spending on the Chief Executive's projects as well.
I bolded the part I somewhat disagree with. The executive is free to negotiate whatever he wants. But, for the treaty to be binding, it needs the advice and consent. A smart, principled executive will certainly take into account what the Senate may or may not agree with. It can then become a political issue.
But, many an executive has negotiated certain provisions, then waited for the right time to present the treaty. There's nothing inherently wrong with that strategy.
Ultimately, the Senate's only role is a yes/no vote on a negotiated treaty.
As for the purse-strings, I totally agree.