I've been kicked off a couple forums for thoughts and replies on this. Sometimes folks get too heated on the subject about the same as oil or tires.
Simple and most direct answer that stirs people up, is just don't let go of the handlebars. But I also know it's a "thing" and we are free to ride how we want. It's a crazy hot topic usually on the FJR and other sport touring forums.
And seems the magical cure over there are the taper steering stem bearings.
I know of 4 Gen 1 bikes that have the same issue. 3 of the 4 have been checked top to bottom / front to back. New tires, new steering stem bearings, wheel bearings, alignment checked, etc. They still wabble once hands off the bars.
I worked for Yamaha for 11 years and had several conversations with engineers. Main point always made was motorcycles are designed with 3 points of contact. Bars, pegs and seat. So when 1 or more are removed, things gonna happen out of the ordinary. Doesn't matter what bike. Forks, frame, weight and bike geometry also plays a part.
Also discussed was that most folks lean back a bit once they let go of the bars, which puts more weight on the rear and lightens the front. Recipe for the wibble wobble.
Similar to the OP, a friend had a 2004 and after he had the forks re sprung, his began to shake once the let go. But once he also addressed the rear shock and it was sprung for his weight and both front and rear sag and such were set properly, his shake disappeared.
Hard one to actually pinpoint and you can chase things for ages and still have the shake. Or maybe change 1 thing and it makes it stop.