The idea of Starship being $10M is that it's fully reusable, so your marginal cost is only fuel/maintenance/launch.
So if a Falcon 9 costs $50M to launch, then 70% of that cost is SpaceX profit/recouping its design cost. Starship is supposed to be even more re-usable so lets assume that they only need 30% of the launch cost for recouping design costs/profit. This would be $87M/launch
So my really aggressive estimate is still almost an order of magnitude higher, which is why I am doubtful. Lets also remember that SpaceX got $396M from NASA for the development of Falcon 9 and Dragon [1]
[0] https://www.elonx.net/how-much-does-it-cost-to-launch-a-reus...
[1] https://en.wikipedia.org/wiki/Falcon_9#Conception_and_fundin...
I'd argue your estimate is not aggressive. SpaceX continues to claim an eventual marginal cost in the high 7 to low 8 figures per launch. I'm inclined to believe that deep analysis over a back-of-the-comment Fermi estimate.
[0] https://www.nasa.gov/pdf/586023main_8-3-11_NAFCOM.pdf