Cut it all out and replaced it with what?
Anything. Writing a script is a long, difficult process. Hard to put me on the spot here, but the first idea I can think off the top of my head would go like this:
If the writer/studio decided Finn has to be in this movie and he has to be a stormtrooper who deserted, why not start the movie with him having already deserted? Do we really need to see him desert? How about we make it a mystery whether he’s bullshiting us or telling us the truth? Make it so Rey doesn’t know if she can really trust this guy or not. You know, kind of like how Han was in the original. Start the film with Rey’s story, which is by far more compelling, and perhaps Jakku is some outer planet, and that’s why Finn flees there and she eventually meets him.
Getting the droid involved and designing a reason to get Rey off Jakku would be another problem for another day.