What would happen if ultralights had mountain bike style treaded tires?
Mountain bike tread is meant to sink into the mud in an attempt to gain traction on firmer ground beneath the mud.  A plane generally doesn't want to sink into the ground (even during braking), if it sinks too far during landing or take-off, the airplane is more likely to catch the forward gear and flip/strike the prop.

Bush plane tires are designed to land on soft fields.  The difference between mountain bike tires and bush plane tires is that the high surface area of the bush plane tires attempt to "float" on top of the soft ground rather than dig into it: