The fact that Sam and Lonnie are the same gender, and having a relationship is the story though. Sam accepting that she's got feelings for someone of the same gender, then pursuing those feelings, then feeling backlash from her parents, from her peers, that's the main plot of Gone Home.
It's what sustains the dramatic tension. It serves the same function as making Romeo and Juliet from rival families. It creates the barrier that must be overcome.
The thing is, this story isn't Sam realizing she's gay - she's known for a long time. She straight up says so, and it's pretty heavily implied that Katie has known for a while as well. Through the Capt. Allegra stories you find that Sam wrote, you see a progression of her realization of her sexuality, as well as her feelings towards her family & authority figures. I think that Lonnie was just her first girlfriend, her first love, and her parents, on top of thinking she was a bad influence, didn't know how to deal with finding out their daughter was gay on top of their own marriage falling apart. It's not necessarily Romeo & Juliet, but it is a tragic love story, even if they end up running away together in the end.
And as far as character motivations being two-dimensional? These are the most god-honest human character motivations I have ever seen. Anyone who has been in any of these situations - a family that is falling apart, falling in love as a teenager with someone your parents don't approve of, knowing that relationship was doomed, having a gay sibling, wanting to run away from home because there was nothing left for you there, any or all of those - will know that these characters acted exactly like any irrational love-struck teenager would in the same situation.