The other method, however seems to be begging the question. They assume that chimps and humans had a common ancestor 6M years ago, then calculate the mutation rate and apply it to humans.
That's not begging the question unless they calculate the date of the chimp-human split using the mitochondrial data, having calculated the rate of mitochondrial mutations given the date of the chimp-human split, which they calculated using the mitochondrial data ...
And since scientists aren't completely stupid, no-one has done that.
So long as they're getting the date of the chimp-human split from somewhere else, such as the fossil record, they are then entitled to use this date to calibrate the mutation rate of ape mtDNA and then use that to calculate the date of mitochondrial Eve.
As it says in the WP article:
A requirement is that the time to the most recent common ancestor(TMRCA) of the sample of lineages must already be known from other independent sources, usually the archeological record.
No circular reasoning there.
Edited by Dr Adequate, : No reason given.