Why? Is nagging/persistence counterproductive?

Yes, nagging is counterproductive.

the full title is "Evidence of Satan in the Modern World", by Leon Christian

Cure of Ars, St. John Vianney also mentions how fasting along with prayer is needed to take care of demons that hold souls.

I suggest the web site of "One More Soul". If the issue is low fertility and you and your wife did not receive info in this area, that site and Thomas Hilgers group.

You can maybe try to get your wife interested in religious topics. There’s many interesting short items online that explain aspects of our faith (like  the Guadalupe tilma, the Shroud of Turin, Eucharistic miracles, Incorruptible Saints, etc.) that she might find interesting. In addition, there’s a very good NO program called Bible in a Year with Father Mike Schmitz, online that goes through the entire Bible over the course of one year. He reads it and comments on the passages read. He tries to give helpful hints as to how we can improve our lives through the examples from the Bible. Very good for conversation about prayer, since many passages are Psalms, etc. Also, you can mix up your prayers by month e.g. This is the month of the Sacred Heart and pray that extra prayer for that month. Anyway, I would try to get her more comfortable and interested in the topics and then attempt to pray with her.