Some do actually use supercomputers to do mathematics.
Most however, thrive in an actual academic environment including spending a lot of time interacting with other mathematicians and going to conferences. There's a big difference between living in an academic environment and spending your whole life thinking about math and being around people who do the same... and working a 9-5 and doing a little math on the side. Not to say it isn't possible, but most reasonable jobs don't leave much left over mental energy for doing maths.
Yeah, and it's hard to get exposure to areas that might be fertile for inquiry if you aren't spending time with other people who are working on mathematics. It's a social networking effect, in the same way that you pick up programming tricks and get exposure to new frameworks/languages/techniques by spending time with other developers.
I'm sure there is a sizable number of "amateur" mathematicians who work on problems in their free time, but it is hard to know what to work on and where to look.
Some do actually use supercomputers to do mathematics.
Most however, thrive in an actual academic environment including spending a lot of time interacting with other mathematicians and going to conferences. There's a big difference between living in an academic environment and spending your whole life thinking about math and being around people who do the same... and working a 9-5 and doing a little math on the side. Not to say it isn't possible, but most reasonable jobs don't leave much left over mental energy for doing maths.