Is this a good idea for a car?
I'll cut to the chase - No, it's a crap idea.
It's not like a car where there's all kind of ridiculous connections that can all go bad and where the chances of 1 thing goin bad out of infinite things is 100%.
You see the majority of the complexity that you so charmingly dismiss as "all kind of ridiculous connections" isn't really about the relatively simple task of steering the vehicle one way or another. Certainly they play their part in ensuring steering accuracy and so on but If you've ever looked at the steering mechanisms on go-karts you'll have seen that just turning the wheels to make the kart go in the chosen direction is simple, much, much simpler than what you'd see on a modern road-car, and that's because a large portion of their job falls under the heading of suspension.
The floors of grocery stores are relatively smooth and level. Roads aren't, they undulate, have potholes, camber etc. Any one who has every tried to wrestle a loaded shopping trolley around a parking lot that's seen better days knows that not only are they a pig to keep going in the direction you want them to but also that the contents are going to be in for a rough ride. That's ok though - your bread and milk don't really mind. Human passengers on the other hand are more likely to start complaining. I've driven one of the aforementioned go-karts on a bumpy track that was much more road-like in it's composition and it was not what you'd call silky smooth, the bruises alone took days to fade.
But let's assume you aren't interested in passenger comfort and for whatever reason you quite like arriving at your destination feeling like you've just gone three rounds with an angry polar bear - your vehicle isn't going to be happy either. You're worried about things "going bad", all those bumps, shocks and rattles are going to put immense strain and wear on car's components, the chassis, the engine, the interior trim, the glass and so on.
Then there's the tires - the front suspension and steering on a road car does a lot to ensure that the contact patch between the tire and the road is maintained, minimizing scrubbing and slippage to keep the wear even, making sure that there is still grip being provided (which is important for both acceleration and braking as well as steering), this isn't a concern for a shopping trolley, they are on smooth dry surfaces the majority of the time and don't have to handle anywhere near the forces a road vehicle does so they can just slap a band of solid rubber or plastic on there and it will easily last the life of the trolley.
Could you make a car that steers the way suggest? Probably - but it wouldn't handle particularly well and assuming you ditched all the suspension and geometry components (the "ridiculous connections") it would be horribly uncomfortable and it would break, frequently and expensively.
There's also the safety aspects - free turning castor wheels (such as the aforementioned trolley wheels) are prone to pulling the vehicle of course in the direction of slopes - a sudden dip away from the direction of travel and you can be all over the shop (har-har!) in a hurry. Additionally your ability to steer the vehicle is going to be highly reliant on having power, sure braking one side and not the other will still turn you but at the cost of being able to do over all braking, and the available turning moment is going to be less than if you can also increase the speed of the other side at the same time. A completely free-wheeling car can still perform some rudimentary maneuvering even in the absence of both drive and braking - something with steering such as you suggest can't, and even worse it's not even going to always carry on in the direction it was going either - instead its going to career along following the contours of the surface it on.
So yeah, it's a crap idea for a car.