Sure, they talk about Mars, and in-space refueling seems radical, but they've yet to succeed at doing anything radical... yet.
Rumor has it they were struggling with the payload fraction w/ the first generation of Starship and they switched to a second generation that struggles with blowing up. A big advantage of the two-stage architecture is that you can develop the two stages independently. Presumably they will eventually get Starship to orbit and bring it home, they will have plenty of time to improve it get the payload fraction up just as they did with F9.
SpaceX took a lot of ideas which had been individually proven before, and then put in the work to perfect them and integrate them in a production ready spacecraft. That is important work and good engineering, but not radical. An aerospike had literally never been flown to orbit at that time (I think still not), so it would have been a way worse fit for the SpaceX method of developing the Falcon 9.
I think SpaceX didn't try to develop aerospike nozzles because the advantages probably aren't that large compared to the mixed nozzle design they are currently using. They also reused the same ceramic heat shield material developed for the space shuttle instead of developing something new.
Compare that to the cancelled "VentureStar": It would have used both linear aerospike engines and a new metallic thermal protection system (TPS) instead of a ceramic one. I remember an interview where Musk answered the question of why they aren't doing aerospikes or metallic heat shields etc, that there are many ways to skin a cat. They are only doing one thing that they think will work, which is not necessarily the best possible solution, but potentially faster or cheaper to develop.
Falcon 9 was a highly competitive rocket without reuse. If they didn't get reuse to work it would have been a successful project. Reuse of the first stage was a huge cost optimization that put it in a class by itself -- but they they did it radically reused risks.
Contrast that to the X-33 which would have required a large number of new technologies to all work to fly at all.
Fixed-cost pricing was also a radical innovation because it drove SpaceX to do everything it could to lower costs. It was known for a long time that reusing (only) the first stage was a good path to lower costs, the SpaceX business model rewarded them for doing it.
SpaceX is highly technically innovative but it's been so successful because technical innovation has been centered around cost reduction and practicality, not chasing high performance for the sake of high performance.
The SpaceX model might need change to get to Mars because of latency. You can launch a Starship to LEO, have it blow up, and launch another one in a few weeks. If a Starship fails to land on Mars, however, you have to wait another two and a half years to try again. Similarly, SpaceX runs everything by remote control from mission control which is great in LEO but to stick a landing on Mars you need something that flies autonomously.
I don't believe the stage landings are remote controlled. I've seen several times where they lost contact with the craft but it landed safely.
It would also be a weird choice because radio connections are way too unreliable to be a single point of failure.