I believe that explanation is incorrect. Impeachment is a political process and not a criminal one. So the Bill of Rights and other safeguards do not apply.
Like it or not, the Constitution gives the House the absolute power to impeach. Benjamin Franklin even once remarked that the House could impeach a president just for being rude.
Now having said that, impeachment should only occur when doing it is what’s best for the country. That’s obviously not the case here.
According to who, you? We are not governed by reports of what Franklin said. We are governed by the Constitution was written and originally understood and intended.
The plain language of the Constitution is clear and unequivocal.
The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.Art. II, Sec. 4.
Now if Franklin actually said that, to be constitutionally accurate he would have to have meant that "being rude" was a "misdemeanor" ("low crime") already passed in Congress as a misdemeanor.
Franklin also commented that if we did not have impeachment, the only way to remove a bad president would be assassination.