The other main purpose of allowing for impeachment and potentially a trial of a sitting president who has fallen under suspicion of abusing his office is that it provides a mechanism by which a president can disprove the charges against him and thus be restored to the public confidence.
If a president has done nothing wrong, but has fallen under suspicion of wrongdoing, he (and in 1787-89 it was presumed that all presidents would be a he) would welcome the opportunity to clear his name. Impeachment proceedings in the House and, if necessary, a trial in the Senate would grant him that name-clearing opportunity.
I said it was quaint. But it has a lovely logic, from a time and place long ago and far away when it was assumed that most members of Congress would be fair-minded honorable folks (and when it was not assumed that national political parties — and the partisan loyalties and logic they produce — would substantially change the question from, “Is he guilty of crimes or misdemeanors?” to “What party am I in?”)
From Madison’s notes on the debate at the Constitutional Convention, Cunningham quotes Ben Franklin, who favored the impeachment and trial provisions because they provided a process for “the regular punishment of the Executive when his misconduct should deserve it and for his honorable acquittal when he should be unjustly accused.” (Franklin also noted that, historically, when no such mechanism existed, the only recourses for a leader suspected of wrongdoing was to kill him or to let him get away with the wrongdoing.)