From the clear wording of Article V, followed by comments from The Federalist, the place to discuss amendments is at the convention.
To continue your clear train of thought and placement from 1800, imagine today if 34 states pass 34 distinct, absolutely unrelated applications. The states meet in convention. Suppose further that the states limited their delegations to consider and vote for only those items submitted in their respective applications. Perhaps the other 16 states don't even show up.
The convention would meet, and quickly determine there wasn't enough support to send amendment proposal(s) back to the states.
The convention would adjourn. Delegates go home. What harm done? None.
It is why the states should meet in convention every year. If there is nothing of collective consequence to consider, so be it.
However, an annual convention of the states would have an enormous effect on Sctous, and the Executive, even if it rarely overturned Scotus decisions or regulatory agency diktats. It would also force a nationwide debate on the nature of government, freedom, and so many other fundamentals that the Left so easily gaffs off today.
The only fly in the ointment I see WRT your proposed annual convention of the states, or any other regularly occurring periodic convention, is that such a convention would probably itself require a constitutional amendment to bring it into constitutional effect, because it would be a major institutional or systematic change that goes straight to the constitutional distribution of powers as between the national government and the States and the people thereof.
Though certainly I am completely in favor of "a nationwide debate on the nature of government, freedom, and so many other fundamentals that the Left so easily gaffs off today," I just don't see how such a thing can be "forced" under our constitutional system of government.