That fear for me is not limited to a constitutional convention.
What prevents the House from impeaching the President or Vice President? Evidence? nope, they make the charge and evaluate the evidence. Legal definitions? No, they are not bound by any legal definition, and can define any action to meet the qualification for impeachment.
The ONLY thing that stops an impeachment is the fear of the majority that performing an impeachment might be politically unpopular -- i.e. they might be voted out of office.
If a vast majority of the people are tired of the president, the house can impeach on majority vote and that majority will probably be re-elected for their efforts.
And if in the Senate the opposing party to the president had a large enough majority, or enough of teh president's party was worried about getting re-elected, they could well find guilt by the 2/3rds necesssary.
We could well impeach a President for "not stopping a war when the people were tired of it", even though that obviously is not a "high crime or misdemeanor".
Of course, up until now, the people we elected to office had some concept of a duty higher than themselves to the country, something the democrats lack. I have no doubt that, if they had a 67-member senate caucus, the FIRST order of business would have been a dual impeachment of the President and Vice President.