If anything, they would stabilize the car, no? Don't bicycles stay upright, in part, due to the gyroscopic effect of the wheels turning? I could be mistaken.
You're right, but for the wrong reason
A bicycle does self-stabilise, and I expect to some extent the wheels on a car do the same, but due to caster rather than gyroscopic forces. On a bike, the front wheel is stabilised by the back wheel - the rigid back wheel lies behind the point of steering, effectively pulling it upright and keeping it stable.
Similar applies in a car. Most are set up with some degree of caster, which helps self-centre the steering. The pivot point of the steering is set ahead of the contact point of the wheel, so as the wheel "drags" along it tries to pull itself in a straight line, and the steering returns to the straight-ahead position.
As for gyroscopic forces on the wheels, the main issue I suspect is that there are already so many other factors working on the wheel that largely overcome the gyroscopic force - you have camber, the aforementioned caster, friction from the road surface, toe angle, and Ackermann angle, which is the mechanism allowing the front inside wheel to turn more than the outside wheel, as it's carving a tighter line.
As has been mentioned, car builders want as little unsprung weight as possible so all these different settings can be more effective. Having heavier wheels/tyres to take advantage of a principle with dubious benefits would be counter-productive.
