Squeezed out is exactly right. The secular elite of the country is at best uncomfortable with, and in most cases actively hostile to, any faith-based organization that is not firmly under their control, or which does not adhere to Liberal dogma.
Given their druthers, and as an ultimate goal, the Catholic Church and all its organs in America would be under the control of a body similar to China’s Patriotic Catholic Association, which full powers to set limits on doctrine, approve Priests and Bishops, etc.
At the very least, they are determined to bring Catholic hospitals and social services to heel — or put them out of business completely.