Racing cars, especially single seaters do indeed have very stiff suspension with relatively little travel, in an F1 car something like 50% of suspension travel comes from the flex of the tyres.
Performance cars are subject to very high loads whenever they accelerate, brake or change direction, ultimately these loads are transferred to the track by the tyres and the amount of load a tyre can transmit before it begins to slide is a function of the vertical load pushing it down onto the track.
A big part of the role of the suspension is to control how these forces are distributed between the tyres. This is further complicated by the fact that cars with down-force generating bodywork are very sensitive to changes in ride height and rake angle, especially in the case of ground-effect down force.
Equally ride height and rake angle can be affected by speed, at higher speeds wings produce more down force and thus more compression the suspension.
Race suspension also supplies adjust-ability no vehicle is perfectly stiff there will always be some flex in the body and tyres but adjustable suspension allows the mechanical behaviors of the chassis to be adjusted very precisely. Bear in mind here that we are talking about very fine margins of performance and complex interactions between, the chassis, tyres and aerodynamics.
It is also important to bear in mind that race tracks are not necessarily perfectly smooth and at racing speeds (the average speed of an F1 car is something like 120 mph) the effect of even modest irregularities in the track surface is greatly magnified. Equally most tracks have substantially raised curbs which the cars need to be able to clip (at least) for best performance.
It is also worth adding that the wheels need to be attached to the car somehow and the loads are the same whether the beams holding them on are moving or not so it's debatable whether there would be any significant weight saving in doing away with suspension. Bearing in mind also that F1 cars usually use torsion springs which are pretty tiny.