You can't really make a living off of being a self-taught mathematician, whereas you can by knowing how to program, and run your own business.
The only way to make a living as a mathematician is to into academia, so that why you see all mathematician in academics. Basically it isn't that academia produces mathematician, although it helps, but that it attracts mathematicians. Just as, say, Y-Combinator attracts good startups.
I would guess the more applied math you care about / the closer to software your math is, the better you can do as a self-taught. I actually know of one guy who transitioned from an engineering ph.d. to teaching applied math doing self-taught / working on fluid dynamics in aerospace, but I expect the examples of someone doing this in more abstract / pure areas are extremely few.
Businesses interface socially with other entities (individuals, governments, corporate customers). That social interface will always be 99% report/client service/cursory analytics and 1% hard science domain expertise. That's just the nature of business.
It could be different in a government lab, a think tank, or a boutique consultancy, and it's definitely different in organizations like prop trading firms that do not interface with outside entities in the way that almost all businesses do. But all of those things also place a much higher value on employing people with significant math maturity and hard science expertise.