BlueCamerlengo, welcome to the CA site. Your post is a good example of some of the differences between so-called liberal and conservative Catholics, which differences gave rise to the question, “Why do liberal Catholics stay in the Church?”
As a conservative (orthodox, small o) Catholic, I would never say, as you do without much explanation that, “After all, the Church is for the world.” Standing alone, that’s far too dangerous a statement; i.e., far too easily manipulated by liberals who --and please note I am not including you–hideously use the Church Herself in attempts to justify homosexuality, contraception, abortion, false social justice, anti-subsidiarity, and even Socialism. You must be aware of the self-described liberal Catholics who are in the news and immediately come to mind because they attempt to justify such sins.
As a conservative Catholic I would say, rather, “The Church is IN the world, but not OF the world.” So long as the world is ruled by Satan*, in this narrow context it would be more correct from an orthodox point of view to say that the Church is against the world rather than for the world.
In closing you say, “If anything, promoting a Universal Church that cares for all peoples should be enough of a reason for a liberal to remain in the Church, even if they find themselves lacking support for the Church.” That, my friend, is the reason this thread exists, viz: If anyone is neither supported by nor can support the Roman Catholic Church, why stay and promote it? Why promote any institution which exits to destroy that which one believes in?
Unfortunately, the truthful answer for far to many is: To continue to subvert the Church. Again, I am not referring to you.
May Our Lord and His “Universal Church, that cares for all peoples”, grant you the answers you seek. You seem at least to know where to look, and that’s a good sign.
*Ephesians 2:2, John 12:31, Colossians 1:13, 2 Timothy 2:26, 1 John 5:19, etc.