Wrong. Couples can opt out of the "automatic state license process" by demanding the priest not submit anything to the state . Also, in many states, it's a separate process. In my state, a couple can get a state marriage license without going through a religious ritual conferring marriage.
The way it works where I am is that a couple can get a state marriage license without a religious ceremony, but it is not possible to have a Catholic wedding without it automatically including a state license. No priest would accede to demands not to submit anything to the state because he would then lose his ability to perform marriages that include a license. Even if one found a priest willing to do that, I doubt his bishop would allow it.
Even if both man and woman wanted a Catholic wedding without a state license, in many (most?) jurisdictions they could not do it. I suspect any claims about women insisting on a state license are based on the "pastor's" experience with non-Catholics. I strongly doubt it applies to Catholics. I have never heard of this being an issue for Catholic couples.