Our Lady of Lourdes

To my knowledge, Fatima and Lourdes are the showcase Catholic miracles. Fatima because it's supposedly well-attested by so many witnesses and Lourdes because it's supposedly well-attested by so many miracles. Not only as Catholic miracles, but miracles that specifically attest the cult of Mary. I've discussed Fatima on several occasions, but Lourdes on fewer occasions. 

In my experience, atheists typically discount the "miracles" of Lourdes as coincidental. Even Catholic authorities only vouch for an infinitesimal fraction of healings compared to the total number of pilgrims.

On the face of it, Lourdes might seem to pose a dilemma for Protestants. If, on the one hand, we accept the atheist explanation, that will boomerang on Christian miracles and biblical miracles generally. If, on the other hand, we credit some miracles at Lourdes, that seems to boomerang against our Protestant position. 

I disagree with the atheist position. It's true that some healings are ambiguous inasmuch as they could either be due to natural or supernatural causes. However, the naturalistic explanation is only available to medical conditions that can naturally improve. But there are medical conditions that don't naturally resolve themselves. 

Conversely, since authenticating a religious claimant is not the only purpose served by miracles, I don't have to rule out Catholic miracles inasmuch as Catholic miracles, per se, don't entail the truth of Catholicism. 

That said, there's a sense in which the issue of coincidence remains, only I'd relocate the coincidence. Even assuming that miracles associated with Lourdes are genuine, yet when only an infinitesimal fraction of pilgrims are healed, in proportion to the vast number who aren't, why attribute a healing to Lourdes? Given the extremely low correlation between healing and a pilgrimage to Lourdes, is there any presumption that the same pilgrims would not be healed if they stayed home and simply prayed to the Father, or Jesus, or Mary, or some saint or another? Is it possible to compare Catholics healed in association with Lourdes to Catholics healed who didn't make the trek to Lourdes? 

Even in Catholic theology, it's not as though a pilgrimage to Lourdes is a necessary condition for miraculous healing. That site didn't exist until the mid-19C. If it was demonstrable that pilgrims to Lourdes experience healing at statistically significant rates higher than Catholics who simply pray to the usual suspects, I don't see how, even on Catholic grounds, Lourdes constitutes evidence for the intercession of Mary.  

A Catholic apologist might counter that even if it doesn't single out Mary, it singles out Catholicism so long as efficacious prayer was directed at the cult of the saints. 

However, that raises another basis of comparison. What about Protestant healings? Protestants who experience miraculous healing by praying to the Father or Jesus?