WP action is not triggering a do_action() call within?

I’ve taken you example and reduced it to this functioning test case:

class ClassOne {
    public static function do_something() {
        do_action( 'my_action_1' );
        do_action( 'my_action_2' );
    }
}

class ClassTwo {
    function __construct() {
        add_action( 'my_action_2', [ $this, 'my_action_2' ] );
        add_action( 'my_action_1', [ $this, 'my_action_1' ] );
    }

    public function my_action_2() {
        print "My action 2";
    }

    public function my_action_1() {
        print "My action 1";
        do_action( 'my_action_2' );
    }
}

$class_two = new ClassTwo();

ClassOne::do_something();

When this code runs, the output is exactly what you would expect:

My action 1My action 2My action 2

So whatever issue there is with your plugin is specific to the actual implementation in your plugin and the other plugin. Firing actions the way you’ve described works fine, as I’ve demonstrated.