I'm not sure this is wrong, but I think it's worth recognizing the two major objections.
1. Capital ownership and opportunity: All of the careers you mention could absorb many more workers in terms of available tasks (something like teaching could easy grow 10x), but that doesn't justify a claim that they will. The farming transition was enabled by mass ownership of productivity gains - careers like entertainment expanded massively because lots of demand was newly available to support that supply. The Picketty school of thought argues that this isn't inevitable; it's possible that a few kids get one on one education, and a bunch of people starve because wealth concentration means there aren't enough buyers for their work.
More specifically, the rich won't buy arbitrary amounts of services because their time and desires are not unbounded. Even 1-on-1 teaching can only employ one teacher-hour per buyer-hour, and the returns on 'background support' don't really justify having 50 people make one lesson plan at living wages.
2. Capability: Take a look at all of the careers you outlined there. They're all skilled-service work, most of which requires having more of some trait than your customer. Teachers need knowledge, counselors need insight, care workers need knowledge or patience and emotional reserves. Crucially, a lot of this is relative - it's not enough for a teacher to know things, they have to have knowledge their customer doesn't have and does desire. Factory labor took some specialized knowledge, but the transition from "a strong back for working fields" to "a strong back for milling cotton" was far less extreme.
This is the sort of two-sided market that's hard to stumble in to. I'm not saying most people can't learn an employable social/service skill, but if you lose your factory job you can't just accept a low wage to work as a teacher. There are significant time and price barriers in that transition, and it's not clear that they'll be overcome without non-market intervention. Markets are wonderfully efficient, but they can certainly decide that it's not worth retraining someone instead of letting them starve.