Bicycle steering geometry is self-righting, the front wheel automatically goes back to center. There is also the matter of flop, which drives the wheels away from center. It's a situation that's ready-made for shimmy. Shimmy is a fundamental feature of steering systems similar to the one used on bikes.
Here are some thought experiments:
--Why don't swing bikes have horrible shimmy problems? They have virtually no stiffness in the direction of shimmy.
-- Did slingshot bikes have shimmy problems? The name of the bikes with the seat attached to a cantilever are escaping me, but it seems like they should also have shimmy problems, since the riders weight is allowed to move sideways.
-- when you get a bad cart that shimmies at the grocery store, do you think they should have used stiffer tubing?
My main theory about why bigger bikes are more likely to shimmy is that they are also more likely to have alignment problems. I really need to strip down a couple of bikes I have that are known to shimmy and see if I can find alignment problems with them.