I think you need a basic course in Civics. First of all, impeachment proceedings are carried out in the House, not the Senate. Secondly, during the trial for conviction in the Senate (after the House has impeached), it is the Chief Justice of the US, not the President of the Senate, who presides.
That’s where you’re wrong, my friend. The Constitution stipulates that the Chief Justice preside over the Senate trial when the President is impeached. That is the lone exception.
The VP was meant to be a backup, with so little actual responsibilty that it never occurred to the Framers he might be impeached.
The Constitution specifies that the president of the Senate preside over impeachment trials, with POTUS the ONLY exception.