Yes, it does move with the steering which works well for a phone or GPS.
A good central location is the Telferizer ball although it rotates with the steering as well.
Are you looking a permanent mount or will a temporary setup do?
Are you looking for a location that can handle both forward and rearward views?
Haven't seen a Gimbal stabilizer on a motorcycle itself, mostly seen it attached to people's helmets and they do say it makes a difference.
This video provides a stark comparison between footage with Gimbal and without.
I have seen some riders with body mount cameras which seem to do well enough. From that, those that upload to YouTube make use of the feature to smoothen out videos from rattles. Useful feature if you ask me.
Past: (not in chronological order)
Diavel Carbon Red
Multistrada Pikes Peak
Owned pretty much all modern Ducati lineup since 2012