Why not allow adults to be adopted as “adult children”?

You can be adopted as an adult. My partner's Grandma was adopted by her longtime step-father in her 50s in the USA. Japan uses adult adoption to pass family owned businesses to nonbiological successors.

Of course the exact rules vary by jurisdiction and country. Here are the rules for California.

https://www.courts.ca.gov/documents/BTB_23_1A_35.pdf

https://en.wikipedia.org/wiki/Adult_adoption

You don't need adoption to get inheritance or personal services. Nor it is a solution to a risky transaction.

Work contracts are always at-will, to avoid slavery. You don't ever want people to be forced to work.

Children don't get inheritance by right. They only get it by default, if the deceased failed to specify a will -- which they can change any time, so long as they are competent.

Agreements of whatever sort to take care of someone for x years in exchange at the end for Y compensation have huge risk that Y will not be there, for whatever reason.

More generally, the main driver for adult adoption is to game the system that privileges in-family transfers of wealth.

The just thing to do would be to remove all benefits associated with family transfers. There's no social benefit to them, and they likely cause intra-familial coercion.

I normally enjoy seeing creative solutions to problems, but seeing that this concept exists is just sad to me and a sign of horrible social incentives.

Outside of very tragic and hopefully rare circumstances, there shouldn't be any question about who helps you out as you age or what you do with the wealth you spent a lifetime working for.