No currently it's not possible to start a routine when someone presses the doorbell.
Please leave feedback for the Google Nest team requesting they add this in the future.
@Cathal_S, thanks for the help!
@Rich1914, Cathal_S is right! We'll take this as feedback as we are always looking for ways to improve our products. We will close this thread within 24hrs and consider it as resolved. You can create another thread if you have other questions. Thank you!