With the front unweighted, if there is no detectable play in any link -not even a millimeter -and the head seings from side to side with no change in resistance, then I doubt it's a problem with any links or ball joints. From my experience with my '07, it's normal for the head to swing side to side with very little resistance even with the damper attached. I think the only other way to rule out the links and joints is to tear them down, which is labor intense/expensive.
Before doing that, consider the possibility that your suspension dampers (front and rear) are worn out. With help, see how quickly the bike returns to a normal attitude after lifting up or pressing down, preferably by comparing to another GT without the symptoms.
Worn shocks on my '07 manifested as a vagueness to the steering that sometimes felt like wandering or even wobbling. I first noticed it at 39000 mi, so it's pretty close to your mileage. I did not notice the usual symptoms of worn shocks such as pogoing, and changing the ESA settings did not affect the vagueness. However, replacing the shocks immediately eliminated the problem.