Connecting the Joystick Shield to the I2C bus won't do anything.
The buttons are digital signals and need to be connected to GPIO pins. The joystick produces analogue signals and the RPi doesn't have any analogue inputs so it needs to be connected to a analogue to digital converter (ADC) chip which can be connected to the I2C or SPI interface.
I used the Funduino Joystick Shield when prototyping my "R-kade Zero" mini arcade and used a MCP3002 ADC
to interface to the analogue joysticks. You'll find a schematic for how I did it on hackaday project page:
In the end I made my own PCB and put it on kickstarter