Why do deep section front wheels get pushed around in crosswinds?
The answer to your title question is "moving air exerts a force on obstacles", and deep section front wheels have more area affected by side winds so there's more force.

In any crosswind there will be some side force that applies a turning torque on the front wheel of almost any bike (viz, zero rake bikes have been made, but they have less trail than is common today). The diagram below shows the normal design, somewhat exxaggerated as was the style in the 1960's:


(via the esteemed Dave Moulton whose post I got the image from contains an interesting discussion about why we have trail)

Note that the centre of drag/centre of the wheel is forward of the pivot axis. Thus, wind force from the side will tend to turn the wheel downwind, in the same direction that the wind is pushing the bike. So the rider will tend to stay upright, unless the turn makes them crash into something.

The bike you have pictured has the opposite design - as the wind pushes the bike and rider away, the drag behind the steering axis will turn the steering into the wind. I predict that in many cases the centre of mass will move downwind while the support moves upwind, resulting in a fall. But note that the front wheel design pictured is at best going to require clean, dry conditions to operate, and those are normally only found on indoor velodromes, making side winds a non-issue.



The OP asks about airfoil effects: More streamlined shapes can generate lateral forces due to airfoil effects, as often helping propel the bike as slowing it down - this is why speed records must be set in zero wind conditions.

Ignoring drag just for this example, the direction of the force from the wind is affected by movement for a streamlined shape, but not for a round shape. The apparent wind direction changes, but the vector components are still just movement + wind. But because drag is nonlinear the total drag force will be greater on the sum (that's why headwinds are so annoying), but the side force will not change by a huge amount.



The key point, though, is that the side force is still present regardless of the shape of the section. You could have a perfectly flat plane and it would still experience the side force.

And what matters to the rider is that the side force exists. Once you have that force, the above discussions of geometry applies. Where the force impacts and how that changes the balance and steering of the bike is what makes the wind hard to deal with. Whether there's a force-aft component to that force doesn't affect balance or handling.

It's definitely possible that turbulence from the fork will mean that there's less lateral force on the wheel behind the fork. But the more streamlined the fork the less true that will be. For the bike you've pictured above there is no effect at all, since the "fork" is merged into the trailing edge of the wheel. Instead that bike will have the opposite effect - there will be more lateral force on the rear of the wheel because there is so much more surface area behind the pivot axis of the wheel (the head tube).