Except, Quintessentially, that means the nanny - quite apart from being expected to break the law in order to receive a reasonable wage - will not have a legal, traceable work record, and if she does not pay NI, etc, it will affect her eventual pension.
I agree that it's done a lot, but - as you say - that doesn't make it right.
And self-employed also means no sick pay, no holiday pay, no notice if the family decide to bugger off to florida for six months, no security at all.
They are utter exploitative shitlords, IMO, but almost certainly think of themselves as naice people providing an excellent opportunity for someone.