You'd think that, but what is really happening is that the plane leads the inertia of your body as it accelerates and then your body catches up. It's a second order effect.
You can observe this when driving a car and doing a hard acceleration - your body sinks into the seat back first, and then once you are up to speed the sensation resolves. The seat doesn't really "push" you. Another way to think of it is that it's not the seat that presses into your back, it's your back that presses into the seat...it's subtle, but it makes a real difference.
I still get to fly this Trainer from time to time (and I've also had a chance to fly the real airplane and can compare), and the way this seat back works is actually annoying to me, once noted and one thinks about how it really should be working. It just plain feels wrong and un-natural!