How to cut gift cards in order to make a mosaic?
I have always found one of the appeals of mosaic to be creating non-uniform / random shapes for your art. Both of these options can help capture that.

Score the card

If you score one surface of the card, with something like a craft knife, you should be able to make it snap along the score line. In general straighter will be easier to snap off. By all means, you can do curve but if they are too extreme or change directions you will have a tougher time removing them. 

Not sure if this will be faster than what you are already doing.

Freeze the card

I know that freezing the card and making it brittle will make it so that you can hit it and it should shatter into easy random shapes. I am not sure if a conventional freezer will get the card cold enough.

I just tried it with my old health card and it did work rather well. It snapped in my hands. However, I handled it too much so it warmed up and started to bend more than break. I would try keeping in the freezer longer. After some time, wrap it in a cloth and quickly hit it again on a hard surface. 

One method to expedite this process would be to wrap the cards inside wet paper towel. Once the water on the towel freezes it should make it easier to process. Again, not sure if this is worth the effort or not.