The first and foremost problem is that you tried to mount a joystick in MDF using regular wood screws.

Wood screws in MDF can not withstand those kind of forces due to the shape/lower surface area of the threads compared to inserts like
these or
these that are designed for use in MDF.

If some players will be throwing their body weight behind the joystick, you might choose to avoid the easy-to-use GGG or Ultimarc inserts and stick with either the carriage bolts or the t-nuts.
There are also t-nuts that have brad holes --
much better for this aplication than hammering the nut to seat the prongs of the other kind.

When top-mounting t-nuts, be sure to countersink them and select proper length screws -- long enough to solidly engage the nut but not so long that they poke through the top when you mount the joystick.
Scott