Church and state was a lot more integrated back then. It was a time when laws were supposed to be consistent with the Bible and lots of activities that were a no-no in the Bible - like money-lending - were forbidden. It's no coincidence that persecution of homosexuals in Western Europe began around the same time as the Western Roman Empire got Christian emperors. Would these laws have existed if sodomy wasn't a sin and the Bible had been silent?
Look at religious persecutions in Europe - the Cathars, the Huguenots, the Inquisition, etc. All of them were done by They were Europeans, but they were also Christians and acting from The goals of the Church - extinguishing heresy, spreading the true faith, etc - and sanctioned by the Church. Why did the Puritans leave England? Religious persecution - by the state, and by the established church in England. Because of European Christians. These actions were carried out in the name of Christianity. You can't separate them by saying "that's not really Cheistianity"
The argument that "homosexuals weren't being persecuted, only people who practiced sodomy" is pretty specious. "I'm not persecuting those people. I'm just persecuting something that a lot of those people do". "I'm not persecuting Christians, I'm just persecuting people who choose to wear a cross" that's obviously persecuting Christians. Similarly "you can live your life - but if you have sex with someone you are attracted to, you'll be put to death" is obviously persecution.
And you dont have to look hard to find Christians talking about Leviticus when justifying their prejudices. Maybe you should ask them why they talk about Leviticus?