> Although I know the church has a lot of money, I can't imagine they dedicate a particularly large budget to the observatory compared to, say, a huge research university or a specialized scientific research project.

The VO has about the same share of the State of Vatican City's budget as NASA does of the US budget (but the Vatican Advanced Technology Telescope has a separate foundation, largely funded by private donations, that funded its construction and funds its ongoing maintenance, without going through the general Vatican City budget.)