Yes, but they are only intended to use it in rare emergencies. It stresses the airframe components and reduces the life of the aircraft. To build an aircraft that lands heavy every time you need a much stronger undercarriage and frame.

and how often are planes fully loaded with fuel? A plane only needs enough fuel to reach the next destination. Saves the weight and money.

Some planes are capable of that and some aren't. Some have fuel dumping systems to get weight down.

https://en.wikipedia.org/wiki/Fuel_dumping

Many are equipped with fuel dumps. Others have mandatory “overweight landing inspections” prior to further revenue flight.

Your car can run past redline, it just shouldn't all the time.