Does someone have a quote of Jesus supporting homosexuality? He defined marriage as between a man and a woman. He had lots of opportunity to speak against bigotry but he didn't.
When it comes to gender and sexuality there is a great deal left unsaid in the Bible.The world lived in by biblical characters was way more complex than simple male and female gender and heterosexuality. For instance, what was an ancient born with ambiguous genitalia supposed to do with a biblical text advising about males doing something obscene with males?
Moreover, and as you seem to suggest, the biblical depiction of Jesus does not have him advocating for civil rights in general. Best I can tell, the biblical Jesus wanted followers who would follow him on his terms. Then in the post-Jesus era someone named Saul came along who had a name change to Paul, and this man changed the face of Jesus followers.
Edited to add: Probably the most common sexual practice among humans is masturbation. Yet the Bible is totally silent about it.