Where is presidential approval in the 17th Amendment or original draft of The Constitution, stating how Senators assume office?
I didn't find where the executive branch has any authority. However,
Article I, Section 5 of the U.S. Constitution states;
"Each House shall be the judge of the elections, returns and qualifications of its own members, and a majority of each shall constitute a quorum to do business; but a smaller number may adjourn from day to day, and may be authorized to compel the attendance of absent members, in such manner, and under such penalties as each House may provide."
This had been interpreted that members of the House of Representatives and of the Senate could refuse to recognize the election or appointment of a new representative or senator for any reason, often political heterodoxy or criminal record. Powell v. McCormack (1969) limited the powers of the Congress to refuse to seat an elected member to when the individual does not meet the specific constitutional requirements of age, citizenship or residency. From the decision by Chief Justice Earl Warren: "Therefore, we hold that, since Adam Clayton Powell, Jr., was duly elected by the voters of the 18th Congressional District of New York and was not ineligible to serve under any provision of the Constitution, the House was without power to exclude him from its membership.".....
The Federal Contested Elections Act of 1969 currently lays out the procedures by which each House determines contested elections.The Federal Contested Elections Act of 1969, also FCEA provides a procedure for candidates to the United States House of Representatives to contest general elections by filing with the Clerk of the House. The law delegates all matters involving contested elections first to the Committee on House Administration, which receives jurisdiction of such matters by the rules of the House..........
https://en.wikipedia.org/wiki/Unseated_members_of_the_United_States_Congresshttps://en.wikipedia.org/wiki/Federal_Contested_Elections_Act