The solid stuff used to make glazing putty is just a filler. It doesn't matter what you mix the linseed oil with. It could be anything that's cheap and easy to grind into a fine powder.

No, that part I fully understand.

Muck dries to form mud. If you get mud wet, it turns back into muck again.
Cement cures to form concrete, but concrete doesn't soften up again if it gets wet.
It's the chemical reaction(s) that occur in concrete (but not in mud) during the "drying" process that make it irreversible.

But, I wouldn't say that "polymerize" is a good word to use here either. In order to have a polymer, you need to have multiple "mers", each of which is chemically identical to all the other mers. In vegetable oils, you have multiple different kinds of fatty acids in every different kind of vegetable oil. Linseed oil, for example contains linolenic acid, linoleic acid, oleic acid, palmitic acid, stearic acid and I expect trace amounts of other fatty acids as well. And, the crosslinking that occurs as it "dries" to form a solid is as likely to happen between unsaturated sites on the same linseed oil molecule as it is between unsaturated sites on the fatty acids of neighboring linseed oil molecules. So, you don't have the repeating pattern that you find in polymers like polyethylene or polypropylene, say. The only chemical group you'll find consistantly repeated a substantial number of times in dried linseed oil would be the oxygen crosslinks themselves, and I'm not sure that would be enough to call dried linseed oil a "polymer".

I think it would be better to say that drying oils "cure" to form a solid when exposed to oxygen cuz the solid they form doesn't consist of the same chemical group(s) repeated over and over and over again in any regular or predictable pattern.