Well, the prots would say that NO miracles are of divine origin (which, come to think about it, is kinda weird). So all “visions” by Catholics are phony, and thus reflect the Saint’s own desire, even if it is a subconscious desire.
But I’m not sure how this can be one of their “biggest arguments” (in your words), because NO Catholic is EVER required to accept ANY post-apostolic revelation. These are ALL considered private revelation, and may be freely accepted (within limits) or rejected (without limits) by any Catholic (which is NOT true of ANYTHING that can be considered the Doctrine of the Church).
If a protestant wants to debate Lourdes or Fatima or whatever, I will say, “OK, fine. Maybe that was bogus. So what?”
The fact that the Church has formally “approved” certain events does NOT mean that the Church says they are true, but only that they are not contrary to the Faith, and the Faithful are free to believe them and teach about them. We may personally reject ALL such teachings, but we are not free to publicly challenge the Church’s approval and claim that the Church was in error in approving [whatever]. We may publicly state our disbelief as our own personal opinion, and we are free to publicly present our own arguments to support our position (and invite discussion), but we may not claim that our opinion is superior to the Church.