I've got a servo set up on my Pi at the moment that is able to push a physical button. Pretty easy setup, add a python script to activate the servo and return to home then call the script in the gpio menu.

Servo: [this one's an overkill but it's what I have] https://www.amazon.com/dp/B07S9XZYN2/ref=cm_sw_r_apan_i_5MSPST9XV4PFND1SHSTS

Servo hat for pi: https://www.amazon.com/dp/B07H9ZTWNC/ref=cm_sw_r_apan_i_65X9PWPMSKPG3687X5CJ