MEXICO CITY — When she walked into an IVF clinic in June, Alin Quintana knew it would be the last time she would try to conceive a child. She had prepared herself spiritually and mentally for the visit: She had traveled to a nearby state to leave prayer ribbons at a shrine dedicated to Mary Undoer of Knots, a manifestation beloved in Latin America. She had received a blessing from her priest. That morning, as Quintana and her husband wound their way for hours from the unpaved outskirts of the Me...