That is an entirely different concept. The sponsorship of art in the country is different than sponsorship of art outside the country. Secondly the sponsorship is done by private individuals instead of a government or military order. CIA cannot be operating in the US and it is a govt institution.
It's not an entirely different concept. The comparison he replied to was about both groups funding art to advance their causes and promote their perspectives (pro-Western, pro-God, etc.). That's a perfectly acceptable comparison and it's accurate. The differences you listed are also valid.
States weren't as rigid in the past. Those Tassis family portraits or whatever cannot be meaningfully separated into "state funded" and "privately funded" because the family's wealth was deeply tied to their relationship with various states.
The church's motivation was precisely to spread Christendom to show the sophistication of the church. So it was very much a work of propaganda used to influence people outside just Italy, but across Europe.
This is not unique. The standard of old used to be Kings sponsoring bards. Both the bankers in Florence and the Church were in essence state actors. The banking families basically ran Florence.
