When chugging ice water, the first few gulps are refreshing, then the cold becomes painful. I’d think, if anything, the cold would have a numbing effect.
Because the longer your tongue/throat is exposed to the cold water the lower its temperature drops.
Per your title: Because ice water is cold.
As for why cold hurts rather than merely numbing, I leave that to those with actual knowledge.
Thermoreceptor nerves are protective, so the brain interprets extremes as pain.
There is a qualitative difference in the sensation from the esophageal receptors, but it’s still basically pain. It would not be “numb” unless the temperature was lowered to the point where the nerves don’t function at all–unlikely from swallowing 32 degree water.