[Lynn Eades]

Even with all that done (wheels and tyres supposedly"perfect") you can still get a shimmy if the hubs are not set up correctly, because any minor play is magnified.
One needs to first accept that any balancing system is limited by its degree of accuracy (in most circumstances it is enough)
If your front axle joints are worn, not centered, you have worn balls in your steering joints or any other un acceptable clearance, then it can quickly magnify into a shimmy.
Just imagine the simplest situation. Imagine a tiny bit of play in a ball joint at the end of your steering arms. When your travelling down the road, the wheels are as per wheel alignment specs. Now just touch the brakes.
The result is that the two front wheels are now trying to retard the forward movement of the veh. The result is that the two front hubs have pivoted back (loading the joints)to a toe out situation. Then when you release the brakes they "flop" back to an "unloaded" position. This is the first part of a shimmy, which can be aggravated by so many things.
Sorting the tyres and wheels often will "fix" the problem, but the "inherent" (tendency to shimmy) problem begins elsewhere.