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.
While I see your point, I would say that if I use a tape measure and measure a man to be six feet tall, and if a second person uses this fact to estimate that he takes size 12 shoes, and if a third person uses that estimate to argue that he can't fit inside a matchbox, this does not add up to circular reasoning. Though it is true that the third person would have made the situation clearer by referring to the original measurement rather than to the estimate derived from it.
This situation would not be altered if matchbox proponents claimed to have a different interpretation of measurements made with tape measures.
But I agree that such dating of mit.-Eve seems unreconciliable with the current evolutionnary-paradigm of human history.
If no-one had ever heard of evolution and everyone believed in fiat creation of species it would still be irreconcilable with archaeology. It's not the ToE that tells us when (for example) Australia was colonized, and there'd be nothing incompatible with the ToE if it had happened last week instead.