Also, are there any helicopters in service that use this mechanism? I was thinking along the lines of using it instead of the part of the swashplate assembly that is used to change the collective pitch.
The purpose of the swashplate isn't to tilt the rotor head. It's to pitch each blade as it passes through various sectors of the disc.
In a Stewart platform, both plates rotate at the same time. In a helicopter swashplate, the control plate stays stationary. Rotor blades pass over the stationary plate and sort of follow its profile in pitch.
Tilting the rotor is not necessarily impossible, but it would not replace the swashplate. Direct motors or servo flaps can do that, at the cost of requiring constant correct computer control.
To replace an effective technology, propose a better technology for that particular application.
The Stewart platform (as shown) would add both weight and complexity, and therefore cost and potential performance limitations, to a function already accomplished with less weight and less complexity. As @Therac pointed out, tilting the rotor is not necessarily impossible, but this mechanism would not replace the swashplate.
Direct motors or servo flaps can do that, at the cost of requiring constantly correcting computer control.
This means that you have to build yet another subsystem to do that: adds weight, adds cost, adds complexity. Granted, complexity itself is not an "all stop" design consideration (or helicopters would not be flying today :) )
Let's consider the next issue: consequences of failure. If you put this system into place, what happens when it fails/breaks? What does the rotor system do, or not do, as a consequence of that? How are the loads being transmitted through the airframe/flight control system in this case?
The Stewart platform is fit for purpose for a variety of applications (one being motion in a flight simulator) but when it comes to helicopter design, you have to earn your way on to the aircraft in terms of the weight required. Until you can explain how this system is better than the system that has been shown to work, what incentive is there to apply that? Given its proposed relationship to the flight controls, which are critical to both function and flight safety, there isn't an incentive readily apparent.
This Wikipedia on Swashplate (aeronautics) indicates
Alternative mechanics to the stationary (outer) swashplate are the hexapod and the universal joint.
So it would seem possible. Very difficult to find anything useful searching for helicopter & hexapod tho, nothing appropriate is coming up, everything seems to be drones, or using the Gough-Stewart platform as the bed for things like testing.
Physik Instrumente (PI) has broad line of Stewart Platform motion controllers, my impression is that these look a lot more complex than a swashplate.
There's this paper on Development of swashplateless helicopter blade pitch control system using the limited angle direct-drive motor (LADDM) which is much more interesting reading.