Adding an admin page – OOP approach not working

I hate OOP mixed in the functional structure of wordpress (here come the downvotes…..). If you use OOP just for name spacing then just use the name spacing feature of PHP, putting function in a class is just a very small part of what OOP should be.

Anyway, your problems is that a there are only two way to refere to an class method, either when it is associated with a specific object or if it is static in the class (basically associated with the class but not with the objects). In your code you try to treat an object bound function as if it is a static one (I am sure that if you will check your error log you will see some relevant error). There are two possible way to fix it
1. Make the relevant functions static and use the syntax in your second variation
2. use a singleton patter, instantiate an object of the dashboard class, keep it as a global and use it instead of the class name in your first variation. instead of

public function setup_dashboard_page() {
    My_Dashboard::add_dashboard_page();
}

do

global $dashboard;
$dashboard = new my_dashboard()
.....
public function setup_dashboard_page() {
   global $dashboard;
    $dashboard->add_dashboard_page();
}

And always debug code with full error reporting and fix all errors and notices 😉