[0]: https://www.budget.gouv.fr/files/uploads/extract/2022/progra...
So you see, when a french person pay for their pension, it's taxes. When a person from another country pay for their pension to private companies, it's GDP. In the latter case, you have the additional freedom of not paying anything (especially if you don't make enough) and then hope that you die before retirement. I would not call this freedom though.
Try to think outside of the conventional wisdom which is completely skewed
When the government pay their pension to a pensioner it is not included in the GDP [0].
[0] https://www.investopedia.com/ask/answers/082415/are-social-s...
Under the income approach, social security tax payments are included in income: https://staffwww.fullcoll.edu/fchan/macro/2gdp_computation.h...
You’re incorrect to say that an economy without social security artificially seems like it has a higher GDP than one with social security. It is counted as income to the employee, and anything purchased with it is counted as consumption.