To the OP, can you further describe the 'jerk' you mentioned that occurred when you went from reverse to forward? What were the exact conditions, going up or down a hill, full stop before changing directions, etc.
Everything fuel related has checked out so far, and it's cranking, but for some reason it won't fire fuel and start. I wonder, out loud, if somehow it's 'confused' as to whether it's in neutral when it is cranking, but it's thinking it's in gear and thus won't allow fuel flow? Something like a faulty sensor/wires touching that are creating a 'fake news' signal in the brain?