Well, impeachment by the House of Commons was invented to PREVENT private citizens whom the King wished to appoint to Ministries from getting appointed. The Founders surely knew this, which is why they included the disqualification option in the power granted to the Senate when it tries "ALL impeachments".
This is a ridiculous jump.
IF this were the purpose of impeachment, then any private citizen could be impeached to simply prevent them from holding office... So whichever party was in power could simply start and impeachment against them, try and convict them and prevent them from running for office...
If this were the purpose of impeachment in the US constitution, then it would be used routinely to prevent candidates.
This is not remotely what the framers had in mind. Impeachment exists as a way to remove an abusive office holder, no more.