Passing path arguments to a form builder
it's clearly impossible to add extra arguments to an inherited function signature


Not true - adding extra arguments to a function signature is one of the fundamental reasons that inheritance exists in OO.

You literally just need to add your arguments to the function signature to make this work:

public function buildForm(array $form, FormStateInterface $form_state, $arg1, $arg2) {
    // You have $arg1 and $arg2 available
}


You could do something more fancy and get typed data by using parameter upcasting, otherwise you'll get the args as strings.

Examples here.