also most undocumented foreigners are undocumented because the procedure to get documented is entirely absurd. For example the Refugee says that asylum applications should take 6 months to be processed, but commonly take around a decade, and sometimes take up to 19 years (1) https://twitter.com/TanveerJeewa/status/1310842296565366785

you also have to apply in person within 5 days of arrival, at a refugee reception office. But Director of General of Home Affairs has closed all but three of these offices since 2011, despite court orders to the contrary. This means that the closest office can be up to 900km away

people are also regularly denied the right to even make an
application unless they pay a bribe, often to both the security at the offices and the administrator, as well as being made to pay fees to translators, despite the Refugees Act
providing that this should be free

and then even if you are successful at obtaining asylum status after all of this, it has to be be renewed every six months. And these are just a few of the constraints for just one of the documentation procedures

undocumented migrants are also paid, on average, less than half of the South African minimum wage, if anyone needed evidence of the ridiculousness of arguments that anyone would choose to do it out of anything but desperation