I would argue that your assertion that "all great art and music" was commissioned by the Catholic Church is false. However, for a long time within that 2000 year span, the Catholic Church did dominate in terms of art and music, although so did many courts and nobles who kept artists and musicians in their households, commissioning the works from them as they would any other servant. Only instead of cleaning linens and making beds, these servants made the pieces of art that adorned walls.
The primary difference between that period and now, though, is a capitalist society and free market economy. A feudal society and economy concentrated wealth in one area (nobility and the Church), and also viewed art and music as an acquired skill, much like masonry or carpentry, rather than a transcendent talent. Art was not produced or available for the masses, only the wealthy, therefore it WASN'T mass produced or meant for mass audiences. Moreover, art was required to be ONLY for the glory of God -- other art work was sinful. The Church regulated art in the same way it regulated everything else, and is not far from attempts to censor and regulate art today (although today, arguably, there is much more debate over what qualifies as "art" and what is just offensive trash.)
I would argue that it has been many hundreds of years, though, since the Catholic Church has had the kind of power and clout to regulate art as an industry, or moreover since the best works of art have been paid for by the Church or even geared toward promoting a message of godliness. Certainly there isn't anyone putting up artists in their homes, commissioning great works for their halls or to dedicate to churches, etc. The great artists of the 19th and 20th centuries were largely people who were completely outside of society, who lived "sinful" lives and did not create works of art for the glory of God by the greatest stretches of the imagination. For the most part, they also railed against mainstream and certainly wealthy society.
How art is made and produced, the economy in which it is produce and consumed, is completely different now than it was 1000 years ago. How can it be at all surprising when the Church, or any church, is unable to commission great works, when by and large, most "artists" today work outside of mainstream, and especially religious, industries?