Saddle valves can be unreliable. More to the point, when you shut it off to install the new fridge, then opened it back up....it may either have not opened all the way, or some debris may have gotten in the line. You will have to start at the inlet of the fridge and work backwards looking for where the flow is slowed down. Of course if you disconnect it at the fridge and have good flow there, then it is inside.
One thought: the fridge has a cooler in the refridgator side which holds a gallon or two of water. It is behind the inside rear panel on the fridge. side. You might just try running the glass filler for several minutes, to make sure all the air is out.