I found it fascinating to read that sentence when I read from the article that routine is critically important. The article didn't seem to say that usefulness or value were what mattered or necessarily considered rather that things were familiar.
More broadly I think that the belief you mention is primarily from the protestant work ethic [1 & 2]. Absent that social norm I'd argue that needing to feel useful and the associated need to work is not part of the fabric of humanity. Feeling needed and feeling that work is a reward is one path to happiness or pleasure but it is not the only one.
I think a case can be made of not wanting to be a burden on a loved one but, sadly & cruelly, at some point dementia robs you of even that worry.
[1]: https://psmag.com/economics/protestant-worth-ethic-real-6554... [2 linked from 1 & more succinct summary]: https://hbr.org/daily-stat/2013/08/there-really-is-such-a-th...