“And Michael Pence is the presiding officer of Congress who rules on how to handle each objection.”
As of 12 noon on January 20, Pence is a private citizen. There would be no Vice President after that point under your scenario.
For most of US history, officers of Congress — the President Pro Tempore of the Senate and the Speaker of the House of Representatives — were first and second in line of succession. Only Trumpies have ever seriously suggested that that was a problem.
In any case, the House can prevent anyone in the cabinet from becoming President by impeaching them. That is in both the 1947 law and the 1886 law that preceded it. Were Pompeo try to pull such a stunt he would probably be immediately impeached.
The real question is why you want America to become an undemocratic fascist dictatorship.