Your article is in error. There has only been one Constitutional Convention, and as per the authorities granted by the U.S. Constitution, that will be the only Constitutional Convention in America's history. What the States are calling for is an Article V Convention. As you explained, the delegates would be given the opportunity to propose amendments, but that is all.
The Constitution cannot be rewritten. As Article V. states, only amendments may be proposed, and then those amendments still have to be ratified by 3/4 of the States. The Constitution cannot be changed without the authority of 3/4 of the States, and changes may only be made through the process of amendments, be it by proposals by the Congress, or by the States at an Article V. Convention. Also note that the only influence the federal government would have on these proceedings, should the call for an Article V. Convention be taken seriously, would be choosing the time and place of the convention. The Congress has no other authorities regarding this in regards to a convention called by the States. However, the U.S. Congress won't call a place and time because, as you may guess, they do not wish to give the States that kind of power.
Also, note that as indicated by Madison's Notes on the U.S. Constitution, originally the States, through a chosen delegation, were going to originally be the only source of amendments, and it wasn't until the second to last day of the convention that they added as an after thought to let Congress propose amendments too. Interestingly enough, all amendments since then have only been proposed by Congress.
So, though I understand your concern that the Constitution could be inadvertently allowed to be rewritten, following the letter of Article V, it is not possible.
Douglas V. Gibbs