“According to our current Constitution, he’s not allowed to hold public office.”
That’s not true. In the case of Alcee Hastings, they removed him from office but did not disqualify him from any other office. The two don’t necessarily go together, it’s up to the Senate at the time.
Again, here's how it's stated in Article I, Section 3, Clause 7: Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States:
I read it as a minumim punishment. I guess the Senate read it as optional.