Not when the Constitution was written. The VP was the one that came in second place in the electoral count. This is how Aaron Burr became Jefferson's VP.It was changed by the 12th amedment in 1804. It seems to me that the impeachment clause was written with that in mind...that the VP likely would be opposed to the President.
It seems to me that the VP only loses the ability to preside in a presidential impeachment and is replaced by the CJ to preside. I don't see where the VP loses his vote in an impeachment as he is still the President of the Senate. The CJ is given a role to preside, but he isn't made the interim President of the Senate...he's still the officer of another branch of government. The Constitution says the President of the Senate (VP) has a vote in ties.
During the impeachment of Andrew Johnson, CJ Chase cast votes and also made rulings. He probably was overreaching on the vote issue, but then Johnson had no Vice President as the office was vacant and no means to fill it.
Because he wasn't sworn in as a jurist in the impeachment trial, the way the Senators were.
Therefore, he can't participate.
-PJ
In your points, it also makes sense that the VP not be a controlling vote in the process - if he’s considered nd opponent, he’s pre-biased.