First, the washing machine lets you substitute one kind of effort for another and thus use comparative advantage.

'Comparative advantage' needs exchanging one kind of effort for another because it's a law regarding trade amongst humans. What you're looking for is mechanical advantage, which often involves trading work for more work.

With an AI, code is code; there's no [mechanical] advantage in writing code to instead of writing code.

No. If you spend 1 day writing code for an AI and then it writes all your code from now on, you've saved an arbitrarily large amount of time writing code.

Second, in referring to "code I could have written myself", I was referring to qualitative advantages rather than time saved.

Well your question was "What's the use of an AI...?", to which I could legitimately bring up all sorts of advantages you hadn't been referring to. If you had said, "What's the use of a chicken when it can't even dance?" I could respond "I could eat it" and presumably that would be an answer to your question.

To make the washing-machine analogy work with this, postulate a washing machine that doesn't actually save you any time - maybe you have to sit about cranking a driving shaft for two hours - but produces much cleaner clothes, or a smaller chance of ripping, or some other qualitative measure.

Your washing machine does still sound preferable to hand-washing, so I'm not sure what the point was.

I'm terribly confused as to what your point was, or why you think a code-writing machine would be useless. I want one!