The wonky wheel on the shopping cart brings us to the matter of suspension. The shopping cart has no suspension. Theoretically all four wheels are mounted in the same plane, and on a flat floor they should carry the same weight. In reality, a tiny twist in the cart, maybe from once being loaded with 14 six-gallon cases of water and three screaming kids jumping on top, will leave the cart largely sitting on three wheels. The fourth wheel barely touches the floor, maybe only part of the time, and the wonky-dance begins.
Or a bearing wears out -- any one of the six bearings (four wheel bearings and two swivel bearings) -- same result.
Take a look at a RhoadesCar -- mass produced in Tennessee. Plenty of them on Playa. There is a tiny rubber "spring" on each front spindle, but it amounts to nothing. So a RhoadesCar travels on three wheels most of the time. The weight of the riders is in the rear, so it's always a front tire that's in the air, which does no harm (but you can feel it in the steering wheel).
The trouble begins when it's a drive tire that comes off the ground -- and the two drive tires are connected by an open differential. The wheel in the air will spin, and the vehicle goes nowhere. Same if one drive wheel is in a dust-dune -- or on snow in Minnesota in January. One wheel spins, vehicle stands still. This is why some cars have a limited-slip differential -- known by many names and also of many designs. Posi-Traction comes to mind. (Never buy a car without it!)
The open diff problem evaporates when you power each wheel independently. In fact, you gain the advantage of being able to aid steering by adjusting the power to each wheel. More power to the outside wheel will help the vehicle turn. (This can be a great help in Kinetic Sculpture Racing, when, for example, making a sharp turn on sand.)
The RhoadesCar has no diff, but it does the opposite of helping the steering. The axle is solid from side to side, but each wheel is mounted on a freewheel, which allows it to spin faster than the riders are powering it. So in a right-hand turn, the left wheel coasts faster, which prevents scrubbing. (Remember that the outside wheel in a turn must travel farther, so it must spin faster.) That may seem fine at first glance. Buuuuut.... Now it's the inside wheel that does all the forward propelling, and... yes, it pushes against the intended direction of turn. RhoadesCars vigorously resist sharp turns. I fixed mine in 20 seconds with a Sawsall -- by cutting the axle in two. Now it has two independent drives.
Adult tricycles, such as are popular in BRC, drive only one rear wheel. Same issue. They turn very well to one side and resist turning to the other side.
Where were we? LOL
No need to limit or connect the swiveling of the four caster wheels in Plan E. They will each take the path of least resistance.
Plan E can also rotate in place, as mentioned when we stuck pins in the kitchen table. The casters will all turn 90 degrees and around we go.
And you can reverse travel-direction. There will be a struggle as the casters fight their way around 180 degrees, but after that all is normal. You may have to label the ends of the platform to know which is which -- though I don't know why you would need to.
The most all-conquering vehicle I have built has four wheels and four riders. Each rider propels his own wheel. Four completely independent drivetrains.
And I didn't even mess with any sort of steering on either axle. I simply built the thing in two halves, and connected them with a pivot in the middle -- like some earth movers and such big equipment. The center pivot can also twist -- that's its "suspension", keeping all four wheels always on the ground. The silly thing is bloody near unstoppable.

Late addition:
Ygmir and Goddammit are right about vehicles insisting on going straight if two wheels on one axle are locked together. You can probably get away with it to some extent on Playa, but on pavement it will absolutely refuse to turn. Beginning Kinetic builders discover this the hard way, almost every time.
There are vehicles that consist of two wheels side by side -- and nothing else. Picture of one of them below. It has two riders, and each drives one wheel. Beginners usually zigzag quite a bit, but after a few minutes they master it remarkably well.
Unwieldy 2 by Dave Hershberger. Photographer unknown.