Article 2 of the constitution states that the president is also

in the second paragraph of Article II, Section 2, that “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.” Thus, treaty making is a power shared between the President and the Senate.

Source: https://constitutioncenter.org/interactive-constitution/articles/article-ii

during the 20th century, the role of the vice president has evolved into more of an executive branch position. now, the vice president is usually seen as an integral part of a president's administration and presides over the senate only on ceremonial occasions or when a tie-breaking vote may be needed.