Very true - I inferred what you thought (incorrectly by the look of it). So if it is now back on topic, I'm interested in the linkage between countersteering and shimmy in your view.
OK, fair question.
First my view on counter steering. At "advanced" riding schools they tell you that you have to counter-steer in order to turn. They tell you to push on the handlebar on the side that you want to turn to. This is told as some sort of revelation and everyone is amazed, forgetting that they've been turning since they first got on a bicycle back in 1964 without ever actively counter steering. Fact is, we don't need to be told to counter-steer. We just steer and it works. Behind the scenes there is counter-steering going on but we can corner as well anyone else without even knowing it. They tell you that "this is how you should steer."
Truth is, counter-steering happens automatically. This is easily proven by riding no-hands. Yes, when you lean into a corner, the steering briefly swivels to the opposite direction, the bike leans, the steering returns and overshoots (the centre) and leans in towards the corner. This all happens without you knowing that it even happens or without you even thinking about it.
That's why I say the whole lore of counter-steering is BS.
Back to shimmy. A shimmy is a severe steering wobble that starts off as an automatic counter-steer and subsequent over-compensation that in itself requires a counter-counter-steer and so on and so on. Bikes which are not torsionally stiff enough to resist the flex and rebound in the front end of the frame do this shimmy at speeds which we can attain. Remember that for each "steer" a whole lot of inertia has to be overcome and it is the stiffness of the frame that has to cope with how the inertia catapults past the centre point to the opposite end of frame flex and bounce back again. The reason the shimmy is speed related is because as you steer, you move sideways and the faster you're going when you steer, the more you move sideways. It is this exact frequency that harmonises with the frame's springiness that brings it on. This flex is more or less about the axis of the top tube but involves the bottom half of the bike moving about your body's anchor on the seat. The plane of the wave is not horizontal though, more like perpendicular to your fork, I guess. Your arse is a node for the wave. Standing up often destroys the node and arrests the shimmy. It also destroys your nerves.
Because intertia and stiffness are involved, we have three variables that makes a shimmy almost impossible to replicate by others going down the same hill at the same speed.
Bikes which are stiffer can also shimmy but at speeds we cannot attain.
Something like a wheel with loose spokes will have a dampening effect o the shimmy, not an accelerating effect. Ditto for loose bearings. A stiff, springy wheel is more likely to contribute than a loose wobby one. A wobbly wheel is highly unlikely to have a wobble that resonates exactly with the critical shimmy frequency. Highly.
Our natural reflexes can induce shimmy. If you have a bike prone to shimmy and you grip the bars tightly and flinch at the onset of a shimmy, you can fuel the fire by your reactions and initiate it strongly.
It is easy to get just about any bike to shimmy. Find a long gentle straight slope where you can cruise downhill and gently pick up speed. Ride no-hands whilst pedaling and sooner or later a shimmy will start. Because you have no hands on the bar, by dropping down and gripping the bar, you can arrest the shimmy. The bike's real hands-on-bar shimmy point is at a speed much faster than that. However, inducing it at low speeds teaches you about onset, control and arrest of this very scary phenomena.
I described my experiments with shimmy here:
https://www.cyclechat.net/threads/speed-wobble.198346/post-4302106