Thank you--I've read the whole thread up to this point waiting to see if anyone talked about phase change, and you did!
You can see this intuitively--when you fill a glass with really cold ice cubes (say, 0F), and pour a little water in, that water will likely freeze--because the ice is well below freezing temp.
But after a while, as the entire glass warms, as you add water it doesn't freeze--the ice just bobs around. The ICE is at 32F but it's the phase change as it melts that keeps cooling the water to 32F.
It's why ice in a cooler lasts so long--takes a lot of heat to melt that ice, which keeps things cool.