> If a polarized cap were used, would it then matter which direction the cap is
An ideal cap: does not matter.
Many types of caps are near-ideal, in the sense that they don't have a DC polarity. Film, paper, glass, mica, plastic, wax-paper: all symmetric. Insulator does not care which side is positive.
The big exception is the Electrolytic.
Have you ever seen electroplating? Copper, gold, chrome? You mix a liquid with a metal-salt, stick in two electrodes, and connect a battery. One of them gets plated with metal. The other does not. Which one is determined by the battery polarity.
To make an electrolytic capacitor: fill an aluminum pot with alkali. Put in an aluminum electrode, preferably foil for maximum area and capacitance. Apply a battery between the pot and the foil. A large DC current flows. If you get the polarity right, a thin film of Aluminum Oxide forms on the foil. AlO is a good insulator, so the current slows to a stop. Now you have a very thin layer of insulating AlO, with a conductor on each side (the foil and the alkali liquid). A Capacitor! And because of the large area and super-thin insulator, a pretty large cap.
This works fine as long as there is DC voltage of the original polarity. Without any DC, the AlO is eventually re-dissolved, though with modern caps that can take many years. With reverse polarity DC, that AlO film is eaten-off very quickly. Now you have no insulator, just a can of conductive liquid. It acts as a resistor, with unpredictable but usually very-low resistance. In power applications, it sucks all available current and bursts. In audio coupling use, a reversed electrolytic will leak enough to throw your amplifier's carefully designed DC bias scheme out the window, and the amp will stop working.
So: short-term, the electro looks like a capacitor even if voltage goes a little in the wrong direction. But if voltage stays wrong, it might as well be a can of onion soup. So you only use elecros in places that have a for-sure DC voltage, and you make sure to put the cap in the right way.