https://en.wikipedia.org/wiki/Independent_scientist#Notable_...
In my own field, fluid dynamics, Robert Kraichnan is very well respected and spent the majority of his career working as an independent scientist. As you suggested, math is cheap, so Kraichnan was a theorist.
This seems to be the largest online forum for independent scientists: https://forum.igdore.org/
Right. One recent example for an independent research in math is Yitang Zhang (https://en.wikipedia.org/wiki/Yitang_Zhang) who, after completing PhD at Purdue University, worked off jobs (accountant, etc.) for 8 years, while doing his research.
Sounds like now is a time for a return to philosophy! Much less capital intensive to be able to make a contribution by thinking, reading and discussing vs setting up a multimillion dollar research lab.
I imagine in mathematics this would be possible, but increasingly less likely as the research becomes more capital intensive