“…which means it determines who gets to go, and when it is”.
Congress does set the place and date, but if they use that authority to unreasonably delay it, the states can go to the courts for redress. But Congress has no power to determine who gets to go to a convention called by the states. The states send their delegates. The Constitution gives Congress its own method of initiating amendments, which Congress controls. A convention called by the states is different.
The Constitution gives the power to call for an amendatory convention using this method to the states, and the Congress “shall” convene the convention; they have no choice. The Founders explicitly gave the people the power, through their state legislatures, to amend the Constitution independently of Congress as a check on a tyrannical federal government. I believe even the state governors cannot interfere, but I could be wrong.
Moreover, Congress does not preside over the convention called by the states, nor does it set the rules or the agenda. The states control their delegates, and several court rulings have concurred; there can be no such thing as “rogue delegates”. In the unlikely event that an unpopular amendment is proposed and accepted by the necessary 2/3 of the delegates, still 3/4 of the states, 38 presently, are needed to ratify the amendment. There is zero danger of a “runaway convention”.
Congress does determine the method to be used to poll the states for ratification of any proposed amendments.