My PhD supervisor cannot promise me a date to defend my dissertation before my visa expires. Should I just quit my PhD and start over?
You should talk immediately with the International Office (or equivalent) at your university, as well as any student-coordinated offices at your university. This is by no means the first time that they will have dealt with such a situation. Usually there are workarounds available—for instance, you could be extended on a part-time contract for a few months while waiting for the defense to be completed. You certainly do not need to be employed full-time to qualify for a visa—you just have to be able to show sufficient income, which even a 50% contract would be enough to satisfy. 

However, I should point out that, depending on the university, there is no way to guarantee that you can turn in your thesis in July and be able to defend it in September. For instance, in the faculty where I used to work, reviewers were given three months to review a thesis. So if you turned in your thesis in July, you probably would not be able to have a defense before the beginning of November. You'll need to check the regulations for your program to see when the defense can be scheduled relative to the thesis being handed in.

Moreover, the "art" of scheduling a thesis defense requires getting enough examiners together at the same time, never an easy task in the German system.

However, to answer your main question, don't you think that quitting and starting over is counterproductive? If you're already writing a thesis, that shows you're relatively near to completion, and that your advisor thinks you're ready to move on. Giving that up because of the logistics of scheduling a PhD defense seems very shortsighted. Why would you not think that the same thing couldn't happen if you started over and worked somewhere else? Since you're so close to finishing, get it done!