I think there are a lot of reasons why this is unlikely to ever happen, not the least of which being the math.
To call a national convention to propose a constitutional amendment would require an application by the legislatures of 34 states. There are currently 31 state legislatures controlled by Republicans (30 where Republicans control both houses, plus Nebraska, whose unicameral, nonpartisan legislature is unofficially controlled by Republicans), 8 states where each house is controlled by a different party, and 11 state legislatures where both houses are controlled by Democrats. That means, assuming all 31 Republican-controlled state legislatures voted to call a convention (which I would not consider a safe assumption), we’d still need at least three states where Democrats control at least one house of the legislature. The good news is that there is at least no language in Article V that would permit state governors a veto in this process—five of the 31 Republican-controlled legislatures have Democrat governors.
It may not be so simple as the legislatures merely voting to call a convention, either. Article V is silent as to the composition of such a convention, the procedures to be used, whether any voting at the convention would be by state or by individual delegate, whether delegates would be bound to follow the instructions of their state legislatures, whether the legislatures can restrict the purpose of the convention, how long the convention would be allowed to sit, or what number of delegates or states are necessary to propose an amendment. There is no precedent to go by, either, because this procedure has never been used in the 227 years since the Constitution was ratified. These are some pretty important issues, so who gets to decide all of this? The legislatures themselves? Congress? What if some of the legislatures’ applications include conditions authorizing only a certain kind of voting or the proposal of certain amendments? What if Congress ignores these conditions in forming the convention?
Assuming the convention is valid and proposes an amendment, that proposal would then have to be ratified by 3/4ths of the state legislatures, or 38 states. That means, again assuming all 31 Republican-controlled legislatures ratify the amendment, ratification either would require at least 7 states where Democrats control at least one house.
As a result, any amendment would have to have very broad support just to be proposed by such a convention, let alone ratified. Which begs the question: If an amendment has broad enough support to be proposed through this national convention procedure, isn’t it likely that it would be proposed by Congress first? This is probably why no convention has ever been called.
There hasn’t been a controversial constitutional amendment proposed since the 1970s—the Equal Rights Amendment in 72 and the D.C. Voting Rights Amendment in 78. Neither was ratified, of course. The last controversial amendment ratified was the 24th Amendment prohibiting poll taxes, which was only controversial in the South. There are only eleven former-Confederate states—two too few to block an amendment.
This is all by design, of course. In rejecting Jefferson’s idea of allowing a convention to be called on the concurrence of two of the three branches of government, Madison warned in Federalist 49: “The danger of disturbing the public tranquillity by interesting too strongly the public passions, is a still more serious objection against a frequent reference of constitutional questions to the decision of the whole society.”
A convention is an interesting idea, but any discussion about it is either academic or wishful thinking at this point. It is certainly no shortcut past the hard work of winning elections and convincing the public to support your views, which is where the focus should be right now.