I mean, this is a bit subtle. With a fixed front fork, it's very difficult to get a bike to lean. It's just the case that with a free front fork, shifting balance may be enough to get the wheel to counter steer, which can initiate a deeper lean.

Having said that, it's 100% the case that a 2 wheeled bike with rounded tires in a lean will turn. Just consider that the circumference of the tire shortens as you approach the edge, it must travel a shorter distance per revolution. so without sliding, it clearly needs to turn.

In-fact, at speed, the lean of a bike is the primary factor that contributes to the turn.