I think by drawing this parallel you are simplifying too much.

There are still horses. Probably more than would breed naturally in the wild. There's less horses now than there used to be because they were bread specifically for their economic utility. Humans are not bread for their economic utility. The amount of humans there are is pretty natural, given the available technology and infrastructure. There's no reason to think there would be less if they contributed less to economic output if the output doesn't decrease.

If you want to compare people to some other animal why not choose dogs or cats? They have no massive economic utility now, event though they had some in the past, yet they still thrive, even better, just because infrastructure and total economical output allows it. Their is even a part of economy dedicated specifically to their needs.

I think being pets of this civilization is something humans might realistically aspire to. I'd say it's immensely better role than being its workhorses.

If you squint hard enough, luckiest of us are already living the lives of pets of the civilization, where they don't have to work at all and the machines made out of money, legal documents, devices and other people take care of their every need, within limits imposed by the machine. None of those people envies the fate of workhorses.

Do you think they define their humanity through the utility of the work they (don't) do?