Chief Justice of the Supreme Court would preside over the impeachment of a President
No, the Chief Justice does not preside over the impeachment of the President. He presides over the trial. Impeachment is the laying of charges, nothing more. If/when the majority of the House votes to impeach, the President will have been impeached. But he still is President, with all Presidential powers until 2/3rds of the Senate votes to convict. This has not happened in the history of the Republic.