It may be worth defrosting the fridge, even though it's frost free. There may be some part that's frozen that's no supposed to be.
I'm trying to work out what action is missing from what you describe. I thought that there were only three actions on an ice maker. 1) Fill with water. 2) Wait for them to freeze, 3) release the ice cubes, often with some electric heating of the ice tray.
Obviously stage 2 is the one that takes most of the time. The fridge-freezers that we've had didn't have any particular device to cool the ice tray. They just relied on the cold air in the freezer compartment. A dedicated ice-maker would probably make more cubes per hour, but the fridge-freezers always made all we needed.