What are the pros and cons of using shielding as opposed to liquid cooling the skin of supersonic aircraft?
I’m not aware of any supersonic aircraft which use ‘shielding’ to protect them from atmospheric friction heating, save for rockets, and a few experimental platforms like the X-15B which used an ablative coating, mostly of cork or rubber, which slowly burnt off during the flight, protecting the aircraft structure underneath.

As for cooling systems, I suspect that the weight of a network of tubes throughout the skin of the aircraft coupled with a heat exchanger and pump to circulate cooling fluid through them was too expensive and complex.  A more efficient method is simply to use the aircraft’s fuel as a heat sink and absorb atmospheric friction heat be slowly warming the fuel in the tanks.  It’s a lighter and simpler solution.

Ceramic protective layers, such as the tiles on the exterior of the space shuttle orbiter can be made fairly light when they are a matrix of nitrogen gas pockets inside of them.  They are quite effective a heat protection on a unit mass basis but can be difficult and expensive to maintain.

Since the 1970s there have been few aircraft programs venturing past Mach 2 so practical applications of heat protection for airframes has been limited.  We will have to see what new developments in military hypersonic vehicles, such as Lockheed’s SR-72 reconnaissance aircraft, are using to address atmospheric heating.