That would be a great argument if there were not so many examples of nations with older demographics, and very much more generous social welfare programs funding their social welfare commitments without vast numbers of undocumented immigrants.
A far simpler example is to examine the impact on the whole economy of undocumented immigrants, whole sectors of the US economy are deeply dependent on undocumented immigrants, removing that dependence would have direct impacts on the viability of many businesses and would drive up costs to consumers.
The dependence on cheep undocumented immigrant workers is a direct result of the pursuit of ever lower costs/higher prices.
There is only one down side to formalizing and managing immigrant worker access, the economy needs workers, the immigrants need work.
That downside is documented immigrant workers would not be subject to the fears that prevent them obtaining better rates of pay and work place benefits.
That would increase costs that would reduce profits and/or drive up consumer prices.
The US economy and consumers is addicted to the cheep costs that come from the use of undocumented immigrants.