I believe the main argument for "backfiring" is that, if you call for a Constitutional Convention (known as an Article V Convention, after the portion of the Constitution that authorizes it), you can't set limitations on it.
I may want it for the purpose of amending the Constitution to read "Corporations are not People, Money is not Speech," or to outlaw PACS and phony "educational" advertising that is purely political.
When my convention starts, if people I don't agree with are organized and motivated, many state delegations could be packed with people who want to alter so "ONLY Corporations And Fetuses are People," "Private gun ownership of any type of Firearm is an Inalienable Right" and "Citizens must Show a Positive Net Worth of at least 500,000 US Dollars in order to be allowed to vote."
Going through the more formal, restricted process of proposing a specific Amendment that must be introduced in Congress, be passed by 2/3 majorities in both houses, and then be ratified by the required number of states (3/4, which would still have to approve amendments from a convention, as well) seems doomed to fail in this current period of hyper-partisanship and gridlock, but the alternative of calling a Constitutional Convention is so broad that it can easily have unexpected results, in the opposite direction that those originally organizing had hoped for.
If you read Article V, it states that Congress can propose amendments, or 2/3 of the states can apply for "a convention for proposing amendments" - this means that the amendments that get proposed and approved will be determined at the conventions. The entire Constitution can be almost rewritten with a few limitations on areas that can't be infringed upon.
US Constitution: Article V