Some of the articles I read about this, including the article referenced in the OP, don't ever mention where that term "seven mountains" comes from.
It is in the book of Revelation's story about the "whore of Babylon" (Rev. 17:1-19:3). The early part of chapter 17 introduces a woman who sits on a beast that has seven heads and ten horns. The woman is the whore of Babylon. Rev. 17: 9 explains that the seven heads are seven mountains (or hills, depending on the translation) that the woman sits on and that they are also seven kings about whom it tells us some additional cryptic stuff.
Several things to notice about this:
1. The author of Revelation only mentions the seven mountains in passing, but he goes on and on and on about the seven kings. The seven mountains are mostly a sidelight in this discussion, not the main point.
2. The New Apostolic Power Freaks (as I prefer to call them) apparently aspire to be the whore of Babylon since she sits on the mountains where NAPFs want to sit.
3. There is no further explanation about the seven mountains. Nothing. Nada. Not a word. The part about "religion, family, education, government, media, business and the arts"--well, that's the part the NAPFs just made up out of thin air, because those are the areas where they lust for power.
(Side note: The lust for power is one of the key aspects of my understanding of Babylon. One of these days, I'll get back to writing more of my perspectives on Revelation in the Christian Liberals and Progressive People of Faith forum, and Babylon is my next intended subject.)