Does anyone know if there are academics (such as mathematicians) who use welfare checks to free themselves up to just do research? That is, as an alternative to the ivory tower, either due to preference or failure to meet selection criteria. How successful have they been without an official post? I feel like this would be an ideal life.
I am not sure what at all is "ideal" about this plan, to be frank. Social welfare recipients usually receive only enough money to ensure subsistence, if even that much. Moreover, in places like the US, there are major time limits placed on the amount of time someone can be on unemployment "insurance" and receive such funds.
So, even if you do get such money, it would almost certainly not be on an ongoing basis, nor would it be a particularly comfortable life, unless you have sufficient additional resources (other working family members) to supplement welfare payments.
Moreover, being unaffiliated with any institution means that you don't have access to the resources of such institutions (journal access, computer and laboratory access, and so on). Obtaining these resources without institutional affiliations can be very expensive, and could completely negate the benefits of being "free of the ivory tower."