I would feel like I had died and gone to heaven if every member in my ward and stake openly and sincerely invited everyone they knew to sit with us as we worship in Sacrament Meeting simply because they love them and want to share our worship with them - no other strings attached.
I would love to see our chapels packed to overflowing during Sacrament Meeting, even if many of the attendees left the building and didn't stay for the other two hours - because they weren't interested in the instruction that occurs in those meetings.
I would love to sit with a gay friend and his partner and their daughter, to smell cigarette smoke residue in the pew behind me, to wave to the girl in the tank top in the back, to see what tattoo or earring the man in front of me had added the previous week, to be surrounded by every shade of skin imaginable, etc - even if I had to wish them a blessed week after Sacrament Meeting ended, and even if I had no realistic hope in them ever being baptized and joining the Church.
In all seriousness, I believe that if we lived the true heart of the Gospel better, these friends would be among us - especially if our efforts to share the spirit of our worship were not tied to "conversion" but were focused more on simple friendship and fellowship - on the joy and spirit and peace of our worship. I think many would accept the occasional talk about the Law of Chastity or the Word of Wisdom or Modesty in Dress, if it wasn't directed in a judgmental way at them and their lifestyle - if they knew our standards didn't change our love for them. (I realize many would not accept it, but I believe many would.)
I have no problem telling the missionaries to stay away from a friend who comes to church with me, if that is what that friend wants. What matters to me is that my friend is there with me.
Of course, I want that friend to accept the Gospel, be baptized and receive the blessings of the Gospel that enrich my life - but that's not a condition of my invitation to worship with me. I wish with all my heart that we could open our arms and embrace anyone who walked through our chapel doors, sincerely and lovingly and unconditionally - and that we brought more diverse people with us through those doors. I don't think we have to compromise our doctrinal standards to do so, but we certainly have to experience a collective mighty change of heart.