Optimal actuator to move an analog stick

vinceherman:
Please confirm or correct my understanding of what you are asking.
You have an existing user input device with a 2 axis joystick and some buttons.
You want to construct a new device that sits on top of that hardware that physically pushes the buttons and joystick to simulate a user.
You suggest that PID will be necessary.

Is that correct?

Absolutely correct. The device would repeat the user action, and this action have been recorded previously in a LOG file with an adjustable sample rate [ms]. I have already done the logging part.

vinceherman:
Do you have access to the UI hardware? Can you snap a picture of it and post it here?
You suggest PID. What will be the input of that PID? Will you have access to the output data from the UI hardware?

I have access to the UI hardware but I absolutely cannot disassemble its part (knob, buttons, etc). Unfortunately I am not allowed to take pictures.
Assuming that I will end up using the X and Y servos, the input of the PID would be the servo positions, the controlled variable would be the X and Y sticks position (I do receive feedback), the set point for every sample time would be the value reported in the LOG.

vinceherman:
It would be feasible to construct your device using RC hobby servos as @PaulS suggests. One servo for each button that needs pressed and 2 for the joystick.

For the buttons I was also evaluating small push pull solenoids. I'll see what's better.
For the XY servos I am so much un-experienced about the linkage / hardware structure. Do you have any reliable guide and shop to retrieve this stuff? Obviously this linkage shall be small enough to manage that stick.

Many thanks for your help!