Why would a chess-playing AI be tuned to do anything except play chess? Just seems like a waste. A bunch of small, specialized AI's seems like a better idea than spending time trying to build a new one.

Maybe less morally challenging, as well. You wouldn't be trying to install "sentience".

It's the other way around - you might want a general-purpose model to learn to play chess because e.g. it improves its ability to reason logically in other cases (which has been claimed for humans pretty much ever since chess was invented).

Considering that training models on code seems to improve their abilities on non-coding tasks in actual testing, this isn't even all that far-fetched. Perhaps that is why GPT-3.5 was specifically trained on chess in the first place.