Can The Vice President Invoke The "Speech or Debate" Clause?

According to reports, former-Vice President Mike Pence will challenge the special counsel's subpoena by invoking the "Speech or Debate" Clause. Politico cites an unnamed source:

Pence allies say he is covered by the constitutional provision that protects congressional officials from legal proceedings related to their work — language known as the "speech or debate" clause. The clause, Pence allies say, legally binds federal prosecutors from compelling Pence to testify about the central components of Smith's investigation. If Pence testifies, they say, it could jeopardize the separation of powers that the Constitution seeks to safeguard.

"He thinks that the 'speech or debate' clause is a core protection for Article I, for the legislature," said one of the two people familiar with Pence's thinking, who spoke on condition of anonymity to discuss his legal strategy. "He feels it really goes to the heart of some separation of powers issues. He feels duty-bound to maintain that protection, even if it means litigating it."

The Speech or Debate Clause appears in Article I, Section 6, along with several other provisions:

The Senators and Representatives shall receive a Compensation for their Services, to be ascertained by Law, and paid out of the Treasury of the United States. They shall in all Cases, except Treason, Felony and Breach of the Peace, be privileged from Arrest during their Attendance at the Session of their respective Houses, and in going to and returning from the same; and for any Speech or Debate in either House, they shall not be questioned in any other Place.

The word "they" refers back to "Senators and Representatives," which appears at the outset of the paragraph. The Vice President is not a "Senator." Not a member of the legislative branch, or something to that affect. "Senator." Text matters. He was, without question, the "President of the Senate." But he was not a "Senator."

The Constitution expressly contrasts the President of the Senate and actual Senators. Article II, Section 1, spells out the role of the Vice President during the joint session.

The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted.

The President of the Senate is apart from the Senators.

Article I, Section 3, Clause 4 provides:

The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.

Senators have votes on all legislation. The Vice President does not.

There is a long-simmering debate about which branch of government the Vice President belongs in, for purposes of the separation of powers. But there is very strong textual evidence that the Vice President is not a "Senator" for purposes of the Speech or Debate Clause.

The bigger surprise is that Pence did not invoke executive privilege. It's possible this "Speech or Debate" gambit may be a not-so-serious effort to fight the subpoena, stand for some institutional prerogative, and eventually give the special counsel everything he wants. (Much of what Pence knows is probably already in his book.) It may not matter much, because Trump will try to invoke executive privilege to block Pence's testimony.

Update: Gravel v. United States (1972) includes this line:

It is true that the Clause itself mentions only 'Senators and Representatives,' but prior cases have plainly not taken a literalistic approach in applying the privilege.

I would hope that the Scaliafied judiciary takes a literal approach to the Constitution.