As someone doing both (well, not sure about the 'successful' part yet =)), I definitely see a lot of parallels in PhD work and startup work, and a correspondingly large amount of overlap between the skills needed to be successful at either.
First there are the 'obvious' primary benefits of doing the PhD: technical skills & knowledge, ability to think about new & important questions, working hard, powering through obstacles of different types, deep focus on a single (perhaps moving) target.
Second there are the less-obvious primary benefits (things that many people don't realize, or that might not apply to everyone). For example, programming skills:
- extreme flexibility: requirements change all the time since you usually don't know what the goal is (or exactly how to get there)
- development speed: a lot of research work (esp. in graphics, which I sort-of work in) requires many many iterations, and the faster you can code each iteration, the faster you can find the true solution
- efficiency: in vision and graphics, you're always dealing with huge amounts of data and processing power is never enough; this forces you to write decently efficient code right from the beginning
Another less-obvious benefit is communication skills. Once you start publishing papers, you'll learn to become adept at many types of communication:
- Writing: Technical papers need to be written well to be accepted. This means simplicity and clarity. Conference papers are severely length-restricted, so you have to learn to get to the essence of a concept quickly and clearly.
- Polishing: Especially at graphics conferences like SIGGRAPH, submissions have to highly polished. This means taking care to get lots of "minor" details right and making everything look very professional, so that it gets accepted. This mirrors the kind of polishing you have to do in startups to make sure customers will want to use your product.
- "Selling": Sometimes viewed as a "dirty word" in academia, there is nevertheless a large component of "selling" required to become really successful in academia (unless you're a super-genius, which you'll quickly find most people are not). This means presenting yourself and your ideas well, making sure to emphasize your contributions and differentiating from "competitors" (previous work). The parallels to startups are obvious.
- Speaking: Presentations at conferences are a big deal -- your chance to make your work widely known and to make a strong impression about yourself to the top people in the field -- people who might one day be your interviewers/bosses/collaborators. The clarity and focus required for good writing is a must here as well.
Third, there are the secondary benefits which many people have already pointed out: "street cred", backup plan, academic "union card", etc.
Being a PhD startup founder, I find that you can tilt the scales in your favor by taking advantage of your domain knowledge. Lots of sufficiently motivated and hard-working people can create successful startups in areas which don't require advanced knowledge, but the field of competitors narrows considerably in areas which are highly technical.
Finally, I'd like to congratulate you on your admission to those PhD programs. They're all great, and I know some people at all of them. Startup-wise, Stanford is probably your best bet, although I know CMU & UW also have had startups in vision/graphics, so it probably won't matter too much. Anyways, best of luck with whatever you choose, and certainly keep your options open -- you never know what direction your interests will take!