Epistemic status: Speculative thinking with a bunch of links to facts I found during my research.
Elon Musk SpaceX already provides launch capabilities for larger satellites that are cheaper then the competition. While it costs $165–220M to buy a launch with the Ariane 5 which can bring 20,000 kg to low-earth orbit, it only costs $62 million to buy a Falcon 9 launch which can carry 22,800 kg to low-earth orbit. That's the quoted price without reuse and rumor goes that SpaceX manages to make 40% profits on each launch.
With Starship Elon Musk is now building a system that's planned to provide another strong improvement in price. $2 million per mission into low-earth orbit. He plans to be able to mass produce Starship at 5 million$ per ship.
Tourism
A Starship is projected to be able to carry a volume of 1,000 m³ with 100,000kg of weight. In contrast a Boeing 737-800 NG which can carry 184 m³ with a weight of 24 000 kg with 189 seats. While it's possible that a Starship can't pack it's passengers as tightly as a Boeing 737-800 NG this still suggests 5,000$ tickets to low-earth orbits that would allow a space tourism industry whose journeys could be afforded by the 1% and not just by people earning millions per year.
Asteroid Mining
When it costs 20$ to transport a kg to low-earth orbit we might find a way to to mine palladium that can be sold for $34,115 per kg on earth or gold that can be sold for $60,882 per kg. While it's unclear how to effectively mine asteroids the transport costs won't hold back businesses anymore from trying to mine asteroids.
Given that the total mining output in the US alone was 692 billion U.S. dollars in 2019 I would be surprised if we don't see a few billions flow per year into the prospect of making asteroid mining viable once Starship is ready.
Tungsten Rods
When Tungsten rods are dropped from space onto earth they manage to store a lot of kinetic energy because they have a very high boiling point. Dropping tungsten rods from space can release as much energy as nuclear weapons without the nuclear fallout.
Starship transport costs will make Tungsten rod cost effective weapons and they should be on the radar of anyone who currently focuses on nuclear containment because they provide similar risks.
EarthNow satellite coverage
EarthNow is a company that plans deliver video that shows Earth from orbit in real time:
Each roughly 200-kilogram (440-pound) satellite would be equipped with a system of four independently steerable telescopic cameras, feeding views into a patented edge processing system that could provide resolution as fine as a meter per pixel. There’d also be a wide-angle imaging system to add context.
“We’re going to be delivering on the order of 20 frames per second, all the time,” Hannigan said.
Constant global surveillance will raise new questions about privacy and how it should be regulated. Even if EarthNow isn't successful I consider it likely that another company will be able to deliver on real time global video.
CloudLab biolab about above the clouds
When it comes to genetic engineering of potentially dangerous organisms biosafety protocols frequently fail. One way to solve the problem is to make all the experiments in low earth orbit where they are far away from earths biosphere and have little risk to mess-up our earth.
Conclusion
Besides the above examples I could think of, it's likely other people will come up with other uses for the new space-faring capabilities provided by Starship and we will see a lot of innovation in space in the coming decade.
That is a reasonable argument, but I think I'm still right: According to wikipedia, the starship's payload to orbit capacity will be 100,000 kg, and the starship by itself, completely empty, weighs 120,000 kg. So it is impossible that the mass of the starship be included in the calculation of payload capacity, even though the starship does reach orbit.
So we can calculate the (optimistic) price per kg to orbit as follows: 100,000 kg per launch of Starship, cost per launch of Starship = cost of fuel + cost of vehicle + maintainance, I remember reading somewhere that the cost of fuel will be around $1M give or take a factor of 2, cost of vehicle is said by OP to be $5M, so basically $0 amortized over even just a few dozen launches... yeah it looks entirely plausible that it could be about $2M per 100,000 kg to orbit, which comes out to $20/kg. And if the price of fuel or maintainence drops it could go even lower.
EDIT: Now I see your calculation above. So fuel costs only $6.59 per kg of payload? That's awesome! It's actually less than $1M! So yeah, the $10/kg figure seems like a reasonable optimistic (i.e. in the long run, after all the kinks are worked out and economies of scale realized) estimate. I think we'll hit it in 15 years, give or take 10.