Totally get why you'd think that, but here's the thing - it's not so much about the cold numbing your face. I tried one of those ice globes last week (weird, right?) and @seanc73 actually mentioned how the temperature contrast helps with puffiness more than anything. The real benefit is the temporary constriction of blood vessels (vaso whatever), which calms down redness for a bit. So it's not just a fancy frozen rock, even if it feels that way at first.
Ever wonder if the cold just makes your face numb so you don't feel the treatment as much? Could be more about the sensation than any real benefit. Just feels fancy because it's from a freezer.