Ok so I did the tests with light aircrafts. It turns out it is actually the wind causing it. I did tests at JFK without wind, and with a strong crosswind. And the veering off happens only in strong crosswind conditions when as soon as the nosewheel touches the ground, the plane veers off towards the wind. While this is realistic, I think it is exaggerated. Why it does not happen right before touching down or when the plane only touches with the back wheels. I can hardly feel the wind but as soon as the nosewheel touches down, the plane turns rather hard.
Is that even realistic?