My understanding is that hands in cold water will lead to blood vessels contracting and thus the entire body not being cooled, just your hands being kinda cold. Their device keeps blood vessels expnaded thus inducing hypothermia.

The diference is between chilling entire body and only your arms.

I've got a ~$120 vest full of a gel that freezes at 50 degrees, so it's not painful to wear. I use it as A/C on my motorcycle here in Florida; but my marathon-running co-worker says he's seen people wearing them for a few hours before races, to lower their core temperature. Looks like it works, but it can be done cheaper.

Do you know why this happens, or how they circumvent it?

Is it because it is water (i.e. not air-cooled)? Or is it because the temperature differential is too great (i.e. dangerously so, according to the body)?

A damp towel might overcome these issues - or wetting your hands and holding them in front of a fan, for evaporative cooling via fake sweat + fake breeze.

They make a partial vacuum in the glove that basically sucks the blood to the surface of the hand where it can pick up the coldness.

If I remember correctly, the glove lowers the air pressure to keep the blood flowing.