So, assuming I am capable of doing research on my own time, should I do a postdoc and then a private sector job or just get a private sector type job?
The big danger for your future academic job prospects is that it's extremely difficult to maintain an impressive research track record in industry (assuming you don't have an industrial research job in which one of your main duties is publishing academic papers). It's not impossible, but many people try and few succeed. The easiest path back to academia is finding a job that aligns closely with your industrial work, for example teaching mathematical finance if you have worked on Wall Street, but even this path is far from guaranteed to work out. Successfully maintaining a research program unrelated to your industrial job and thereby landing a job at a research university years later is exceptionally rare, so you should recognize that there's a real risk that you won't be able to get such a job.
Eight years after your Ph.D., I doubt anyone really cares whether you did a postdoc first or went straight to industry, so the credential itself is unlikely to be important. On the other hand, the experience and connections could be invaluable in developing and maintaining your research program. The reason postdoctoral positions are so important is that almost all new Ph.D.s can benefit from more research experience and mentoring. The chances that you have truly reached your full potential in graduate school are slim, and you'll be competing against people who have the benefit of postdoctoral experience. There's no reason you have to do a postdoc, but not doing one may put you at a serious disadvantage.
On the other hand, I have also heard the argument that people get PostDocs because "they are not quite ready to be on their own yet". Or PostDocs are used to gain more contacts and learn what it is to be a "professor".
This is not completely wrong, but it makes a postdoctoral position sound too much like remedial education. Instead, even the most brilliant and successful mathematics students typically do postdocs, which help prepare them for even greater success.
As such, any high paying, senior type position would look better than a PostDoc. Even an internship would look better than a PostDoc. Right?
For a typical mathematics hiring committee, no industrial position will be considered relevant at all unless it's directly connected to the candidate's research or teaching. Anything else will basically fall into a black hole entitled "time spent away from mathematics". Being an executive might look slightly more impressive than being an intern, but neither one will help you get a faculty job.
Overall, I think the most important factor you need to consider is how much you care about keeping your academic options open. If this is a secondary consideration, and industrial jobs look more attractive to you, then going straight to industry could be your best bet. If you really want to be a professor someday and would be crushed if it didn't work out, and if the industrial jobs you have in mind aren't particularly relevant to your academic plans, then you should think hard about whether spending six to eight years in industry is the right approach.
