I made a similar set up. I found the motion to be jerky and abrupt, so I've wanted to introduce some delays to smooth it all out. Another problem is that the kit returns to its starting position when you let go of the joystick. I might try to freeze it in place and only return to the origin when the button is pressed. I got mine to operate a laser pointer, with the button controlling the on/off on the pointer and the servos controlling the motion. What I'd like to do next is to integrate this with another project, a two-axis solar tracker, and use the button to switch between manual (joystick) and automatic (tracking) modes. There are plenty of uses for the button, each with similar coding problems and solutions. Cheers.
Nice work, I hope you were able to make much progress. For the jerky motion, it can sometimes be solved if you separate the joystick power supply from the motor power supply (the motors tend to cause noise on their power supply). Also Alan G. Smith (author of IntroToArduino book) suggests that we allow for a "dead zone" around the middle of the joystick where there tends to be much fluctuations.
for those interested, you can buy these parts from china for half or a quarter of the price on ebay if you dont mind waiting a few weeks for the shipping (which is free most of the time)
Those servos are 4 for $5 on eBay smh people really need to look around before just ordering from the first site they look at lmao and that's been the price for them on eBay for the past ten years
They also need to realize these are very use specific items that require specific servos to work and assemble properly. They'll forever be a PITA unless the servos are centered before assembly. These were engineered for giving motion to first person view cameras, and mounted to radio control airplanes especially but can also be used on R/C cars, trucks, boats,etc. This guy is painful to listen to. If you are going to put an informational video on RU-vid, know what the fuck you are doing!