The traditional Christian view of same-sex activity derives from more direct statements made by Paul, and Jesus’s general statements about sexual immorality. What Paul actually meant about homosexuality depends on ambiguous translations from the original Greek. Moreover, Jesus’s prohibition on sexual immorality depends on what counts as sexual immorality. Christian denominations that accept same-sex marriage interpret these statements differently. This is particularly easy in Protestant Christianity because it accepts that the Bible is the work of people and that it must be interpreted.
On one hand, I think you’re correct that a lot of this is results-driven rationalization—looking at the Bible anew with the desire to reach a particular end goal. That in and of itself isn’t necessarily that big of a deal, though. We do the same thing in the law, with texts that are far less ancient and where the original meaning is far more apparent. The Supreme Court just decided this year that Congress had already banned discrimination based on sexual orientation and gender identity back in the 1960s.
But there have also been theologically relevant changes. It wasn’t until the 21st century that it became widely accepted as a matter of science and sociology that homosexuality is innate and not a choice. These facts are enormously important theologically. Christianity is not in the business of affirming peoples’ choices. But an overriding principle is that God created each person individually and knows and loves each person. Within that framework, an orientation that is innate is very different from behavior that is a choice. Moreover, an overriding purpose of Biblical marriage is to have and raise children. In the United States, large scale studies showing that huge numbers of gay couples were raising children together were released in 2011, based on 2010 census data. This went largely unnoticed among folks who don’t see child rearing as an essential purpose of marriage and life generally, but is tremendously important for deciding that same-sex unions fall within Christian (and other traditional) understandings of what is a marriage.
On sexuality being innate/not a choice. The Bible does not prohibit any innate characteristics, it prohibits actions. Actions are always choices. Homosexual actions are what the bible prohibits, the Old Testament first and then repeated (and quoted) in the New Testament.
Marriage is not involved in these prohibitions.
I don't see it just from reading this verse (and knowing the section).