I don't think the "you let Rin die" was the reason why he's helping Madara per se, but the reason why he didn't come back to the village even though he survived.
We're going to get more reasons, or at least one. Not that I'm expecting them to be much better, but still... can't be much worse than "you let Rin die".
"Hello students, today I got lost on the road of life." -Kakashi