While I'm not necessarily comfortable with the apparent intertwining of church and state on marriage, I think we actually do a pretty good job of keeping the two marriages separate already. For example, conservative Christians typically hold that divorcing someone and marrying someone else is immoral, unless the divorce was for a valid Biblical reason. There's nothing stopping a divorced person from getting a marriage license from the government, but some churches will choose not to recognize it, and may choose not to allow that person into their congregation. People (and churches) right now are perfectly free to say "Yeah, but that's not a real
marriage" to remarried divorcees, and in the future they ought to be just as free to deny the "realness" of same-sex marriages, without impacting the legal benefits of civil marriage.
My dad is divorced from his first wife, and as far as I know, it wasn't for a Biblically valid reason in the eyes of most conservative Christians. Further, I was born out of wedlock, and my mother did not marry my biological father (in violation of Leviticus). If the conservative doctrinal definition of marriage were legally binding in its entirety (which some
would like to see happen), then their marriage would be invalid, and our family wouldn't have qualified for all the legal benefits and tax breaks that marriage affords, and who knows where we'd be now. If my parents can be legally married despite not fitting every Christian's definition of marriage, same-sex couples ought to be too.
As for Christian arguments supporting same-sex relations, I prefer Justin Lee
's arguments. Matthew Vines's video is good too, but I think he does make a few arguments that are a bit specious. Ben Adam
has a somewhat more liberal treatment that focuses more indepth on the historical background of Rome, and A Letter To Louise
makes some interesting points that I haven't seen much elsewhere.