OK, there’s a lot to comment on, so I suspect this is going to be a long post.
I am aware that large amounts of caster generate “shimmy”, but I don’t understand the physics involved. Perhaps someone can help here?
But you can get “shimmy” if the vehicle is toed out. The vehicle can (not always) “hunt” back and forth, trying to find which tire is the dominant one. This doesn’t occur if the tiersds are toed in.
Wheel end vibrations (out of balance and out of round tires and wheels as well as non-concentric mounting) tend to show up at the wheel hop frequency - which occurs in the 50 to 70 mph range. For those who are engineers and physicists, the wheel hop frequency is just a manifestation of the resonant frequency of a spring/mass/damper system. That’s why you will hear comments about these things being speed related.
These vibrations tend to be in the radial direction (with reference to the tire and wheel), but if you reference the vehicle, we are talking vertical - up and down. Nevertheless, this vibration results in a steering wheel the vibrates side to side - which is commonly misinterpreted as “shimmy”.
Vehicles are not very sensitive to vibrations in the lateral (side to side) direction. So when you hear folks point to a rotating tire that ‘wobbles’, that may be a red herring. Plus there’s a stiffness of the tire variation that contributes to vibrations that isn’t referenced when we talk about “out of round” - which is why you’ll hear vehicle and tire engineers talk about “uniformity” which includes both.
While I am not a big fan of low profile tires, I don’t think they cause as many problems as they have been accussed of.
And lastly, internet searches: Finding complaints about a product, company, etc, is easy using search engines. That’s what they do. But it is important to find comparable complaints - and… ah … compare. What you may think is a huge problem, might be a small one in comparison.