As most on this site can probably attest to, one of the biggest issues with part-time/hobby/side projects is motivation and discipline. That problem seems to be already partially solved by the freedom to pick an area of mathematics one is already interested in.
Not having enough time to devote to a problem part-time doesn't seem like a large problem if somebody wants to work on problems of their own choosing as per the OP. Thus, my confusion.
I can definitely see that collaboration is a tougher nut to crack unless most participation starts shifting to evening group meetings.
The New Yorker had an interesting article about Yitang Zhang and his discovery concerning bounded primes. The discovery itself is interesting for the mathematics involved, but of wider interest is that up until this discovery, he was a relative unknown in the mathematical establishment.
If it were simply a question of motivation and discipline to have math as a hobby, we would expect discoveries from unknown individuals more frequently, but, alas, that's not true, which is what makes this story precious.
It stems from the fact, among other things, that math isn't like programming. It's not like deciding, one day, that you want to write your own operating system, and then the small matter of coding it up. Math requires* those leaps of intuition that come to you after banging your head against the same problem for months. Most employers would start to question a developer's continued employment if it took them five months to resolve a single ticket.
*http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
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.
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.
I think there are two non exclusive ways of doing research successfully. Intense work and collaboration (in particular supervision of students).
PhD students and postdoc fall in the first category. They work full time on their research and do little else.
On the other hand, most successful experienced researchers are busy writing grant proposals, teaching, attending various committees. There's not enough time left to do meaningful research by themselves (plus getting older doesn't help to stay sharp and passionate), so they end up supervising students that to the bulk of the research.
I think someone that would try to do research on its spare time won't have enough time to devote to it (unlike a phd student), and won't be able to supervise students (unlike a professor).