I don't see why it couldn't be done in a crunch if we had to for some reason. SpaceX isn't pursuing such a thing because it would be a dumb idea rather than wait for the Starship program.
Falcon 9 regularly sends 17 Tons to LEO 2-3 times per week and is human rated. We could probably build a Mars shuttle in orbit, similar to how the ISS was constructed. We've already landed an SUV-sized robot on Mars. There are some open questions like orbital propellent transfer but it doesn't seem like it would be impossible to me.
I believe they already have the propellant transfer worked out. There was a paper a few years back looking into Musk's general plan. They already had some mechanism for refueling. If I remember correctly, the biggest questions were around lifesupport systems with their efficiency and heat shields. Which were technically still solvable with current technologies and slightly different approaches.
From my perspective, the thing they're mostly waiting for is additional tech to make everything easier and lower risk, or for an aggressive administration that can push it and fund it.
Falcon 9 regularly sends 17 Tons to LEO 2-3 times per week and is human rated. We could probably build a Mars shuttle in orbit, similar to how the ISS was constructed. We've already landed an SUV-sized robot on Mars. There are some open questions like orbital propellent transfer but it doesn't seem like it would be impossible to me.