The author completely forgot to include the thirty year war, it was instrumental in completely breaking Habsburg hegemony "almost" overnight on the historical scale, and opened way for historical shift of power to Western Europe from Central, and Eastern Europe.
The Thirty Years War also confirmed that the primary legitimate basis for power would be noble birth instead of ordination by God - a shift begun by the Reformation.
The French Revolution/Napoleonic Wars began the shift away from noble birth and WW I&II confirmed that the primary legitimate basis for power would be ownership and property.
No. (See: Chinese Mandate of Heaven; the multiple examples of people claiming to be ordained by gods and similar higher powers historically despite no or little connection to nobility, so on.)
The system it backs up is heritable monarchy, it didn't precede it.
Assuming this is true (which seems highly dubious), what are you contesting? Their claim is that God ordained their rulership, not that 'local religious authorities' did. Regardless of existence of the entity in question, "the primary legitimate basis for power would be noble birth instead of ordination by God" is the right thing to say.
Seems trivially true, to me. The european feudal system developed out of the late roman imperial system, that would have vicars and dukes, with the dukes representing secular military authority.
Also, off the top of my head, I think somebody like Charlemagne (as the first guy I know who merged military rule with the support of the pope) would have been wholy interested in christian support for secular reasons (and vice versa). Being the roman emperor, appointed by the pope, would presumably help with his legitimacy in Italy and Gaul.