It isn't "rigid moralism" for the Church to be clear about what constitutes behaviours and attitudes that are unacceptable before God. And it's not an insistence on "outward perfection". Catholics believe that once we are justified before God in Baptism, aided by grace, we embark on a process of internal sanctification and seek to conform our lives ever more closely to Christ. We also believe God's grace can be lost through persistent sin. 

It's good you are acknowledging that LGBT people are "hurting, afraid ... often alone" and "broken". All of us are sinners and everyone in search of support from a parish community and grace through the proper reception of the sacraments, are very welcome. As I understand 12 step fellowships, people who attend acknowledge their weaknesses and are seeking to amend their lives. Can this be said for all LGBT persons? Don't many want the Church to change her teachings and positively endorse their behaviours.