The very first vice president, John Adams, thought that the role was bigger than that and he was rudely disabused and nearly shunned by the Senators whenever he sought to lecture them or to master them in any way. That reality has obtained to this day. That is why vice president Texas Jack Garner famously said, "the vice presidency isn't worth a pitcher of warm piss." The vice president has no constitutional role other than to cast a vote in the Senate in the event of a tie, and to keep his heart beating one heartbeat away from the presidency.
Historically, presidents have shunted their vice presidents away into powerless corners and only rarely does a personality like Dick Cheney exert substantial influence and or even construct a power center within an administration. However, although every vice presidential candidate makes a solemn vow to his running mate that he will support the administration and every modern vice president has done so, the vice president remains a constitutional officer. That means he is constitutionally and legally independent of the president and need not support the president and can't be fired by the present. It is only party politics and the word of the men that keeps vice presidents on the reservation.
As to "mastering the Senate" that has rarely been accomplished in a body in which controlling the Senate has been compared to herding cats. It is a body with 100 egos all of whom think that they should be president rather than the man who happens to occupy that office. It takes a political genius like Lyndon Johnson to "master" the Senate and he did that as a senator, not when he was vice president.
I see no reason why Ted Cruz would surrender his office in the Senate to become vice president except as an office to hold place to run for president himself in 4 or 8 years and he can do that nearly has well from the Senate. While vice president, he will find himself in the same position as Joe Biden or Hillary Clinton, tainted with the record of the president and the record of a loose cannon in the example of President Donald Trump. Cruz is a doctrinaire conservative, Trump is an ego on the loose with no philosophical ties to conservatism. Cruz would be crazy to submit to the kind of administration that a narcissist is likely to produce. If he did become vice president and the Trump administration proceeds as erratically as I predict, Cruz would have to violate his verbal pledge and associate himself to the lunacy or exercise his independence as a constitutional officer and openly run off the reservation and repudiate Trump's follies.
It makes no sense for Cruz to accept a vice president opportunity from Trump. Likewise, Cruz would be foolish to offer the office to Trump because Trump's ego is uncontrollable and he would undermine sound conservative governance being advanced by President Cruz.
