Two things stand out for me about this cartoon. First, when the preacher says "God loves everyone!" and expands on it by saying "Gay, lesbian, bi, transgender" without including "straight", he is speaking from a place of privilege. Privilege here assumes that "of course God loves heterosexuals; what is not clear is whether or not God loves those who are non-heterosexual." Inherent in this type of privilege is the assumption that the privileged majority is included, is right, is best, and so on, and usually those who are part of it never question or even think about such things.
Secondly, are gay and lesbian and bisexual and transgender people not from families? Do they not have families? And are there not families in the church who have gay sons and lesbian sisters and bisexual cousins and transgender partners? So the not-yet-spoken implication by the pastor that "because this is a family-friendly church we can't have LGBT people running around", is actually counter-family, or family-unfriendly.
If Jesus had been as picky as we often are about whom we associate with, he would have avoided the lepers, the woman at the well, the woman caught in adultery, the woman who had been bleeding for years and years, the beggars on the street, the man who was demon-possessed, Zaccheus and other tax collectors, prostitutes, and general sinners of all sorts. He would have spent his time with nice religious people, reading the Torah, and looking good. And with all of that he wouldn't be Jesus anymore ... he'd be a Christian*.
* north american variety. And yes, I realize this is a grand generalization which is not fair to the many north america-based followers of Jesus who do get it. However, this is what the majority of non-Christians perceive when they think of Christians.