I've seen this point from several so-called 'reactionary feminists' recently (major ones are Mary Harrington, Louise Perry, Nina Power- basically former radical feminists who now keep some of their old ideas but argue that women need to return to traditional sexual, childcare etc norms at least to some extent), especially Mary Harrington.
I think it's over-simplistic and unhelpful on several counts. For one, an woman employed as a nanny isn't necessarily a low-income woman forced into it by economic necessity. Some, eg Norland Nannies enter it specifically because they're interested and are often from financially stable backgrounds with other options open. Obviously Norland Nannies aren't the typical nanny but it feels patronising for a supposed 'feminist' to ignore that some women do want to do this job.
There definitely is an issue with women being employed as nannies from poorer countries like the Philippines and exploited- paid unfair wages, sometimes physically abused even. I read Ben Judah's This Is London recently and he had a whole chapter of interviews with Filipina Nannies which was very eye-opening on this.
But I think it's more difficult to blanketly dismiss any poor woman working as an overseas nanny as being exploited by a middle-class woman. If a poor woman has few options where she lives, with little prospect of government improvement (which is the case in some places) and can make money to potentially give her children a better education/life etc by working as a nanny overseas, it's a terrible situation- but is she necessarily being exploited if her employer pays her fairly and treats her well?
Calling difficult choices wrong and unfair is one thing, but exploitation seems a bit strong to apply to all cases, and also in a sense patronising to women. If a poor man made difficult choices to try to improve his family's situation by working overseas, would an employer who treated him fairly be automatically termed exploitative?
I suppose maybe these women like Harrington (who herself says she uses childcare) are trying to make a more structural point that women's work often depends on poorer women needing to work as Nannies due to economic problems, even if the women individually treat their Nannies well generally. I'm still doubtful though as that ignores the large number of Nannies who aren't poor and/or overseas workers.