Michael's answer is good - there isn't enough space for a full size nut.
But this leads to the next question.
Instead of having special wrenches why not just use normal nuts and get the spacing from somewhere else?
You'd have to make the hub flanges narrower or the drop outs wider and the axle longer. Everything is a trade-off.
If you make the flanges narrower you would reduce the lateral stiffness of the wheel.
Most front wheels are stiffer than similar rear wheels. Structurally, this is because front hub flanges are typically wider than rear hub flanges. Rear hub-flange spacing is constrained by industry-standard dimensions, such as cassette width, dropout spacing and symmetrical frames.
Wheel Stiffness Test
If you keep the hub flange width the same the axle would need to be longer and the drop out wider.
I'm probably saying this wrong (my physics friends please correct me)
The distance from the bearing to the drop out is longer meaning greater leverage for the forces transferred back and forth between the wheel and the frame.
The thinner cone and nut allows the axle to be as short as possible, the drop outs are closer together and the distance from the bearing to the drop out is reduced.
Thin cones and nuts also have benefits in frame design.
A bicycle is a system with a multitude of design trade-offs that impact one another.
Many rear wheels have spacers on the drive side that dramatically increase the axle length on one side in order to accommodate sprocket clusters. That being said, a thick cone to accommodate a normal wrench would only make it worse.