Have never experienced this, and have had it flat out a few times. Been blown about by wind tho a few times but that's not at all the same thing.
Is bike new or second-hand?
Grasping at straws, possible ideas are play in swingarm pivots, fork clamps and/or front wheel spindle loose, or wear in the sliders, so forks are twisting/flexing, that kind of thing. (I have experienced this on a Moto Guzzi Spada years ago, it caused suicidal high-speed weaving!)
If bike is second-hand, has it been in a previous accident of some sort? Has anyone disturbed the suspension / steering for any reason?
Re tyre pressures, OK, but is your pressure gauge accurate? I bought a new one few month ago and found I'd possibly been keeping all my tyres too soft! Also it confirmed that my 40 years old pocket type 'pop-out' gauge was probably still accurate after all.
Try ducking down low yourself or sitting taller, further back or forward as it may be it's you that is the aerodynamic problem.
Possible. The classic cure for stopping a high speed weave was lying down forwards (aka, lying on the tank, but not applicable to a scooter!)
Also does the X10 scoop air into the fairing anywhere except the radiator, and has it got ways to vent this pressure sufficient at high speeds into a head wind? Could be something simple.
There are two very small vents on either side of the fairing and below the headlamps unit, but purely for decoration as far as I can see.
There is a gap below front edge of screen, where the Piaggio badge is, so that air blows up the inside of the screen as well as the outside, presumably to reduce turbulence around the rider's head. The standard* screen itself is not too large in area and quite narrow so can't see how that could be causing the problem TBH.
* which raises the question, is the screen the factory standard one or something different?
