The key to my getting my Ph.D. was research, research, and research. The first I did was on a little question that came up in a course. In two weeks I got a surprisingly good answer that was clearly publishable. Later I did publish, no problem, in JOTA. That work gave me a halo in the department and made the rest easier.
For my dissertation, I had identified the problem and made good, intuitive progress on it before I entered grad school. In my first summer I turned the intuitive stuff into math, theorems and proofs, wrote an 80 page manuscript, and that was the research for my dissertation. So, I found my own problem, found my own solution, did the work with no direction, and submitted the final work. So, in no meaningful sense did I have any dissertation advisors. So, I didn't have to depend on an advisor.
Myself I'm not particularly worried about maximising income or whatever. I like what I'm studying and researching, and I'm paid enough to live a comfortable life at least for now. I could honestly do this for the rest of my life (which is what an academia career is, I suppose), but if I ever get fed up with it I also know I have loads of doors open everywhere, which is certainly a comforting thought.
Best of luck!
I'd say TCS and pure math are about equivalent in terms of the career opportunties they offer.
Something I forgot to mention: under no circumstances should you self-fund a PhD. Make sure you have a stipend to keep food on the table. Self-funding magnifies the risks I've described tenfold.
This is a realistic perspective of what grad school is like in the average-to-bad case. I could identify with parts of it:
http://nautil.us/issue/56/perspective/what-does-any-of-this-...
Second this. I was told before starting "If you have to pay to do a PhD you're doing it wrong." This may not apply to fields like Literature, but it most certainly applies to Math and CS.
It's a risky choice, unless you're very good. You should go in with your eyes open, knowing that you'll be lucky to get an academic job afterwards, especially if you're in a 'pure' field without many applications. For industrial jobs, you'll probably have to re-train through self-study. After a maths phd, doing so may come easily, but it will also take effort.
You'll spend three to seven years (dependent on your location and ability) working on an extremely specialised topic. There is no guarantee that you'll finish: circumstances beyond your control like your relationship with your advisor can derail your progress regardless of your ability. There is also a 'point of no return' after a year or two where it becomes very painful to cut your losses and walk away with nothing.
Doing a maths PhD could open some interesting doors. It will improve your employability, but it's an inefficient way to do so. The only good reason to do one is that you really like your research topic and want to spend years immersing yourself in it.