That had never occurred to me but is a great question. I suppose between the poor conductivity of the fabric and the paint inside the dryer, the system is very well insulated. Maybe a dryer with bare walls could be antistatic without the use of dryer sheets. But maybe that would be a fire hazard.
As the clothes get dryer they become less conductive. Even if there was a perfect path to ground, the charge wouldn't be able to get there across the material itself.
That can't be the whole story though. You can feel the static charge from the clothes on your hand when you reach into the dryer a good 6-12" before you even make contact with the clothes.
the psu is grounded, but the static has no way of getting to ground (via psu) if the case itself is non-conductive.