@ondrej @mjg59
Okay so I am not a climate scientist or oceanographer, just a physicist, so take this with the commensurate amount of sea salt.
You're right that the salinity of ocean water lowers its freezing point. This isn't by much - about 2°C - but that can be a big deal given the high specific heat capacity of water.
However, once water freezes it can stay frozen for a very long time. (Think of icebergs drifting out of polar regions.) Ice melts from the outside in, so the less surface area the ice has relative to its volume, the longer it takes to melt. You may have observed this if you've seen how long snow takes to melt compared to how long ice takes, or compared the melting times of ice cubes in a drink versus crushed ice.
(Mathematically: the volume of ice increases as the third power of scale, while the surface area increases as the second power, so the larger a lump of solid ice is, the longer it'll take to melt, assuming constant external temperature.)
The bottom of the ocean is very cold. More importantly it's very dark. There's not a lot of energy there, apart from a few volcanic vents scattered around. Convection currents mean that whatever heat there is will rise away from it, and the very coldest water will descend. A large lump of ice down there would last a very long time.
So why doesn't it? Because as soon as water freezes down there, it becomes ice, which is less dense; this makes it rise even though it's cold. Eventually all the ice collects on the surface where it's exposed to the warmest water and to sunlight, so it'll melt. An ocean is an ice-melting machine.
Let's hypothesise a world in which water didn't expand when it froze. The coldest part of the ocean would then freeze first, which is the depths. These would become single huge homogenous lumps of ice, fed by convection currents. If the world ever got temporarily cold, the entire bottom part of the ocean would just be a solid mass of ice.
The world did get that cold, around 100,000 years ago, and only warmed up again about 12,000 years ago.
Twelve thousand years wouldn't be enough time to melt such an immense lump of ice, especially once enough surface water melted to shield it from direct sunlight. Most of the ocean would still be frozen, which would have all sorts of knockon effects that I am not an expert on. For example, it would concentrate the salt in the upper (liquid) portion, making it so saline that it'd kill any fish.