What’s good for society is not so much for the church.
Subsidiarity rejects all forms of tyranny. It makes hierarchy more a matter of enabling those in the middle and bottom to carry on their lives than giving those at the top the power to plan out what is wanted and see to its achievement. It rejects the conception of social justice most common today, which emphasizes equality and universality and thus a comprehensive system of supervision and control. Instead, it stands for the Catholic and classical conception of social justice, a state of affairs in which each part of the social order receives its due so it can carry out its proper function.
More generally, it rejects present-day liberalism, the attempt to turn the social order into a technically rational contrivance for maximum equal satisfaction of individual preferences. It opposes it not only in its leftist or progressive form, which emphasizes expertise and equality, and prefers to act through neutral bureaucracies and international authorities, but also in its rightist or conservative form, which emphasizes energy and efficiency, and prefers global markets and the exercise of national power. So it is ill at ease with both the politically-correct welfare state and such aspects of present-day capitalism as outsourcing, big box stores, the penetration of commercial relations into all aspects of life, and the bottom line as the final standard for business decisions.
It nonetheless accepts certain tendencies often identified as conservative or liberal. It generally favors family values, distributed powers, federalism, local control, and freedom of enterprise and association, all of which now count as conservative causes. It also favors causes that count as liberal, such as grassroots democracy, limitations on big business as well as big government, and certain kinds of unionism. It favors neighborliness and an active civil society, which everyone says he likes, and maintenance of borders and limits on globalization, which our major parties along with the whole of our ruling class now reject.
The life of the Church provides a concrete example of why subsidiarity makes sense and how it works. The point of the formal structure of the Church, her hierarchy, sacraments, disciplines, and subordinate bodies, is to help the faithful become what God intended them to be. That purpose can’t be legislated, administered, or forced on anyone, but it can be aided, and that is the point of what the Church does as an organized community. As the saying goes, salus animarum suprema lex (“the salvation of souls is the supreme law”).
To that end, the aspects of the life of the Church that normally matter most—parish life, the availability of the sacraments, and the religious life of the believer and his network of family and friends—are necessarily local.
I do not understand why papal supremacy and the centralization of church power in Rome fits neatly with such a subsidiarist outlook. The Reformation questioned the centralization of European church life. Protestants are inherently subsidiarist while Roman Catholics have to hedge.