I ended up using https://eu.switch-bot.com/products/switchbot-bot -- just have a finger robot push the button

Hah, I did the same exact thing and came here to say that :) I was looking at wiring diagrams and telling myself I could wire up some arduino circuits for it but gave up when when I realized I could just press the button!

edit: although mine was an ancient system from the early 90s. It was just replaced with a modern system a couple months ago. At my previous apartment I had wanted to set up a system that would allow either my then partner or I to activate the callbox and have it set for a VOIP number since we could only put one number on the box.