Now to try to find out why it doesn't output anything to the port.
Well, that's easy. When is your class constructor called? When is the HardwareSerial constructor called? You don't know, do you? Therefore, you should not write code that relies on them being called in the correct order.
You need a begin() method in your class. You pass nothing to the constructor. You pass a serial instance to the begin method. You call the begin method in the setup() function.