And yet in the original comment you never said it would be a gay love story. You just said gay story.

But let me rephrase that then - would you call a love story between a heterosexual man and a woman a “hetero love story”?

The “gay” in “gay love story” is a secondary classificator - it’s a love story of two people who happen to be gay. A “gay story” could be a love story, but could also be a story of someone finding their identity, about their struggles in the world as a gay person etc.