The requirements for amending the current Constittution would be moot if the Convention drafted a new Constitution with different ratification requirements. As noted, it would not be the first time as that is exactly how our current Constitution came to be.
There is no provision, except for amendments, in Article V, for the wholesale re-write of the U.S. Constitution. The language is very clear. And at a convention, two-thirds of the states have to approve any amendment for it to be sent to the states for ratification. Each state has one vote, and it would only take sixteen states to keep an amendment from getting approved in the convention, and sent to the states for ratification.
And the amendment process, proposal and adoption, is not a popular vote. It remains a vote of the states, with each state having equal voice in amendments. For those who think that a New York or California would be able to highjack the process, remember, that for every New York or California, there are North Dakota, Wyoming, Idaho, Texas, and any other number of states who believe in the original document.
And what happens if, the states which believe in the strength of the Bill of Rights, decide they're not going along with any amendments which diminish those rights? It would only take thirteen states to send any amendments down to defeat, if those amendments even got approved at a convention. Right off hand, I can think of more than thirteen.
New York, Los Angeles, and Washington, D.C. may be the center of the media, but flyover country is where the Constitution lives, and is believed in by the country's citizens.