Is my plan for sticking with my incompetent advisor reasonable?
What are the chances I could actually get a postdoc?


Of course there's no way to give a definitive answer without far more information, but here are some factors to consider.

First, how bad do you think your thesis would really be?  It's possible to graduate with a thoroughly unimpressive thesis, good enough to qualify for a degree on the basis of hard work but not good enough to satisfy any employer who cares about research.  If that's your situation, then the chances of a worthwhile postdoctoral position are negligible.  On the other hand, perhaps your embarrassment is because you're writing a merely average or even good thesis rather than a truly noteworthy one.  If that's the case, then you might be fine.  Perhaps the truth is somewhere in between these extremes, but we can't know where based on your question.

Have you had any feedback from third parties?  For example, have you spoken to other mathematicians at conferences?  What about visitors to your university or collaborators of your advisor's?  If other mathematicians have heard of you and expressed seemingly genuine interest in your work, then that's a plus.  If you've never interacted with anyone but your advisor, then you're in a much worse position to look for a postdoc.

What about letters of recommendation?  You'll need several from people other than your advisor who can address your research accomplishments.  Do you have letter writers in mind?  If you can't think of anyone, then that's a major obstacle.  If you can, then have you sounded them out on whether graduating is a good idea?  If they are unenthusiastic about the idea, then you aren't likely to get compelling letters from them (and without such letters you won't get a decent job).  If they are as enthusiastic as your advisor, then you should feel reassured.

Does your advisor have a track record of helping students get jobs?  If so, then that's an excellent sign.  If he has a track record of failing to help students get jobs, then that's a terrible sign.  If he doesn't have a track record at all, then that's worrisome, and his views on the job market might not be well informed.


  Assuming I obtain a postdoc, is it likely I could establish a decent track record of research, given that I will be in some sense two years behind schedule as far as learning this field and establishing myself?


If you can get a three-year postdoc, you'll be able to accomplish a lot in that period.  You may not have time to achieve your full potential before going on the job market again, but you'll be in a much better position than you are now.  If you get this chance, then it's worth taking it.

Incidentally, I don't think it's fruitful to count how many years you are behind schedule.  There really aren't schedules or mileposts for research, and different people's research programs and abilities often develop in incomparable ways.  Some mathematicians do their best work much later than others, for reasons that are not always apparent.


  Staying an extra year with this advisor may be an option, but would be difficult financially. 


Perhaps you could apply for jobs now but fall back on staying another year if your job search is unsuccessful.  That's not ideal, but it might be your best option.  If you really can't afford to stay another year, then your choices are far more constrained.  You either apply for jobs this year or choose another career.