Why does wax pool under a candle wick, and why is wax even used on a candle?

Wax is the fuel that is being consumed.

Some wax melts and forms a pool at the base of the burning wick - we say "burning" wick but it isn't the wick that is burning, it is the wax liquifying, then being drawn up the wick and vaporized by the heat of the flame, that is doing the actual burning.

Why, indeed. Without the wax, you would be attempting to burn a plain piece of string, which wouldn't last long.

Candle wax is a hydrocarbon fuel, just like oil or gasoline. It happens to be solid at room temperatures.

The candle flame melts the wax, which is drawn up into the wick by capillary action, and burned.