If the states agree on a "general" convention, that's what would occur. But only three times in our history, all by states who were unhappy with the results from Philadelphia, have states sought a general convention.
All other conventions must be specific to the agreed subject. And a call for an amendment to get rid of the UN does not match in subject matter a call for a balanced budget amendment. And Congress is not obligated under Article v to call such a convention until 34 states have agreed on the purpose of that convention.
No such agreement has ever been reached. The article you cite ducks this question, which is decisive.
Thanks for the clarification Congressman Billybob.