Sorry, you're right that the pitch or a worn cog is changed, but it's only changed because the shark-finned ramped shape of worn teeth forces the chain further away from the axis of the cog or chainring, thus increasing the distance needed to wrap a given number of teeth on the cog, exerting force on the chain to lengthen it. I'd figured that out in the second part of my post above, but should have gone back and changed the first post.
This mechanism also explains why larger cogs (let alone chainrings) are less likely to pose a problem either for skipping or wearing out a new chain.