The process for the precaria and DNI that you link to sounds like it was meant for Mercosur residents, even though they did have a note of what it costs for non-Mercosur residents to apply. I saw that because the process seems the same for anyone who would get their residency, which is what you need to work legally here as a foreigner, no matter what class of work you do.
So my point would be, anyone from Mercosur countries can already get a DNI and they want Mercosur citizens who come here be residents and be paid in the white, and of course anyone else from other countries as well, but they don't seem to mention what program the non-Mercosur residents would be using the get their residency. I don't remember there being a program for domestic workers, for example, where there is rentista, spousal, work sponsored by local employers, etc. Maybe there is, but if not I don't think the au pair from Norway would have a way to get residency.
I could be wrong about this, obviously, but most Argentines who hire maids or aupairs here, for example, are unlikely to be hiring US citizens (or Norwegian, Canadian, etc) to make them residents but rather Paraguayans, Bolivians, Peruvians, etc (who, as nearly as I can tell from knowing many of these nationalities who works as live-in maids, for example). While anyone can apply for a precaria and onward to residency, I think they still have to have a specific program under which to qualify for residency and be successful at obtaining it.
I can understand your friends' concerns about bringing someone with them who may not be legal to stay and work here. Personally, it doesn't seem to be too problematic at this point in time, although that can change. But for anyone who comes here and stays without residency, I would NOT suggest that they leave specifically to "renew their visa" anyway (due to the multitude of conversations held on this subject). I think the person would be alright coming in and working with the family without a residency (and being paid in the black, of course), as long as they weren't planning to leave and return to Argentina frequently. But I can also see where they might not want to take that chance.