The argument isn’t that AI brings the labor cost down to 0 in isolation. It brings the labor cost of the same amount of production down. So you get more production (more things = more supply -> lower prices) out of less labor.
We don't need "AI" to figure out that technological advances increase productivity. The problem with yout argument is assuming that increased productivity mean overall reduced costs. It does not.
Healthcare, housing, education all have gone up despite increased productivity. Then you have things that are already so automated that there is no way to make them cheaper unless sacrificing quality - food, clothing, etc.
Then we have all the types of consumer products that have prices completely decoupled from the cost of labor. No one in their right mind the "cost of labor" has any relation whatsoever with Apple charging $1000 for an iPhone and/or Motorola charging $180 for a Moto G.
Healthcare, housing, education all have gone up despite increased productivity.
The hypothesis of Baumol’s Cost Disease is that these industries are exactly where we should expect prices to rise because they’re still dependent on low-productivity-growth human labor.
We were talking about infrastructure costs under increasing labor productivity. Now what are we talking about?
If the premise is that AI won’t improve productivity in industries like healthcare, education, and housing construction, then why are we worried about “the dead economy”?
No. You are getting it backwards. The premise is: even if AI improves productivity, we the people are not going to benefit from it.
The mistake you are making is that you are assuming that a system where productivity per unit of labor is higher automatically translates into increased global output. It does not. This idea of a dead economy theory is precisely the concern we are heading to a world where machines can make practically everything on the cheap, but it won't matter because the moneyed class won't need to satisfy the demands of the general populace.
Its also the 'labor theory of value'. That's the economic theory that Communism is based on. It has never been accurate and wasn't even considered legitimate during Marx's lifetime. It has possibly the worst track record of predictions of any theory ever conceived by people. Yet somehow academics still reference it. Nobody who actually is impacted by making the wrong economic predictions does though. Funny that...