If you want an actual button pusher for educational/entertainment needs, then you can ignore this answer.
Given your description, I would assume that this physical button is a normally open push button (NOPB) with two wires going to it. In this case, I would think you could come up with a simpler solution that involves tapping one or both of those wires. For instance:
If one of the wires is a common wire that goes to ground, a Pi output pin configured to ground the other wire might be sufficient (with grounds bonded between the Pi and the device in question). If neither wire goes to ground, but I am still correct about the NOPB, it should still be simpler to control some other sort of normally open contact between those two wires in parallel with the button so that a human can still press the button, but the Pi simply briefly connects the wires together without the button.