I would like to convert this to a joystick control with button to drop the claw. A countdown timer would be great as well.
Possible?
It
might be possible, depending on your skill set.

Seems like a conversion like that would involve replacing the game board, changing the control panel/controls/wiring, maybe replacing the gantry assembly (the motors have to work with the game board), and maybe adding a timer display.
Several years ago b0dyr0ck2006 built
this crane game from a kit.
A kit like that might be a relatively easy way to do the conversion.
Scott