Punishment of impeachment isn’t directly being disqualified from public office, that’s a portion that has to be added and voted on by the Senate as a whole as well. I read some story sometime ago about a federal judge who was impeached by the Senate but was able to run for another federal office because the Senate never added the disqualification portion.
Here’s a source on that:
Unlike removal, disqualification from office is a discretionary judgment, and there is no explicit constitutional linkage to the two-thirds vote on conviction. Although an argument can be made that disqualification should nonetheless require a two-thirds vote,855 the Senate has determined that disqualification may be accomplished by a simple majority vote.856
But yeah, I’ll agree with the fact that Trumps defense was 100% bullshit. Nothing more and nothing less. I understand why the turnover rate was so high and why this guy was a last minute choice.....