I was thinking about this today (God only knows why, but I was). I think I might have the answer. It's a tyre problem.
I know what you're thinking, wtf. No seriously, I think this is what it is: a (or both fronts) tyre has de-laminated internally, the steel belts and the nylon belts, etc, have de-bonded inside. You could look at the tyre and even examine it and never know. But if you got it up to a specific speed all hell would break loose because of the centrifugal forces.
There are a few causes for this, but sometimes it's just bad luck. It's quite rare, but common if a car has been sitting a very long time without moving much. Heat and time do it, so does a manufacturer fault.
A free way of figuring this out would be to swap fronts to rear, and rear to front. OR, borrowing some replacement wheels.