At a high level, pedigree of an institution (professional or academic) should not be used as support for a GMAT score. Whatever capital you have in the pedigree will be wasted in that effort. So you do have pedigree on your side, so why use it up here?
The unfortunate truth is that Stanford, like aerien mentioned, is incredibly competitive. Their average GMAT is the HIGHEST of any top 50 ranked program*.
So two things. For one, definitely diversify and apply to other programs. Secondly, retake your GMAT and shoot for at least a 720, if not higher. A 700 puts you in play for sure (and it's a great score) but if you're looking at Stanford, you need higher.
*US News & World Report rankings
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