Wider tyres: increase mechanical grip, allowing a car behind to follow the car in front, closer.
Narrower wings: reduce aerodynamic downforce, once again, allowing the car behind to follow the car in front, closely.
The 2 must happen together. If only one happens, the designers will develop the car to such a point that the reduction in downforce will be negligable and you will be back to the situation where most of the grip is aero and not mechanical.
If the rules banned the aero add ons (winglets, barge boards etc) then the designers wouldn't be able to get the aerodynamic grip to compensate for the loss of mechanical grip. This way the car shapes will be simpler & the airflow won't get so disturbed & as such it will enable cars to follow closer.
Ultimately there has to be a balance between mechanical grip & aero grip. The cars in the 70's etc didn't have the added winglets or always have massive tyres yet the racing was probably the best of any era.
The closest forms of motorsport are always the ones where the cars are most similar & of simple aerodynamic designs.