Your children are argentines with or without DNI. Do the Opcion de nacionalidad at RENAPER. Immigration law is clear that it does not apply to children at school.
Hou was un part of the class action for apartheid back on 2011 where we acchieved to change the rules: https://lndnoticias.com.ar/2011/05/20/recurso-de-amparo-que-solicita-suspension-del-campeonato-metropolitano-de-tango/
As soon as you do not show up at Immigration to apply for a renewal, you can stay for ever. I had a client who was here 25 years before she applied for citizenship.
You do not become illegal.
You pay a small fine when you leave.
The advantage is you do not get a deportation order and you do not need to lie because since you apply for citizenship you cannot overstay as an undocumented citizen.
The law bans to charge More for age, so they have a full fee and deduct age. Hospital Aleman is a good option but crowded. Medical attention is better in the country side, specially public, in places like San Luis where they have low population.
Nowadays 50% of the Federal Judges of the first hierarchy already accepts applications without the 2 years and they do not make you wait until the 2nd year that is a requirement of permanent residency. The fastest case I won this way was 2 years plus 52 days.