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The thing you're missing is the fundamental way a four-wheeled vehicle turns--and the role of caster in it. Just turning the wheels one direction or another will not affect turning of the vehicle. It will 'push like a dump truck', as circle track boys say. You have to tilt the front wheels in order to jack weight, allowing the load to be lifted from the inside rear wheel. The wider the tire, the wider the patch, and the longer the 'arm' formed by the width of the tire outside the pivot axis of the front tire. This magnifies the weight-jacking affect of the castor of the steering, jacking more weight and thus affecting more turning of the vehicle per degree of steering wheel travel, thus feeling heavier as you turn that wheel. Moving the front wheels out towards the fender lips (like a 25mm wheel offset) has the same effect, although to a greater degree.
And such, it's the same case on both points. The area matters not, it's the geometry of the entire system. You can't simplify the geometry of the contact patch down to area, because the area isn't involved in this phenomenon. It's the change in width, translating into a change in weight-jacking, translating into a change in feel and effort at the steering wheel. You further lessen the weight-jacking effect by using a tire with a more rounded edge (higher profile). The edge is not as effectively forced into the ground by the castor of the steering, posing less resistance to the turn of the steering wheel.
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