Plastic typically oxidizes, or turns yellow, and becomes brittle with age. So, why are old music LPs still playable decades later?

Well, it does not turn yellow, because it is black.
And who is to say that it does not become brittle?
You should test it, and break some of your old records, and compare it to breaking new ones.
Also dubious that brittleness would affect playability.

For one, there are different kinds of plastic and they decay at different rates and for different reasons. It’s UV light causing most of the damage though, not oxidation. LPs are typically stored in sleeves out of light and will last decades, if not centuries when properly stored. The yellowing you see on ABS plastics (not vinyl) like old toys and stuff is semi-reversible, look into “retro brighting”

Old vinyl records – say, pre 1960 – were made thicker, and of some other less flexible substance. I have seen them break with barely any pressure on them.

Sometime in the 50s or 60s, the vinyl changed.

@elbanditoroso
Imagine a world where this technology had won against the record.

I have many decades-old plastic items, as do my family and people I know, who have even more older plastic things. Most of them are still intact and as functional as ever, even after decades of regular use, which generally has much more impact than simple age.

Users of Atari computers from the 1980s have found that direct sunlight (in moderation, of course – you don’t want to melt anything) tends to remove the yellowing effects their plastic can experience over many years.