Falcon 9 reentry is only designed for first stages, and the reentry burn is pretty minor, on the order of a few hundred meters/second. A second stage, reentering with far more velocity than a first stage, can't shed the extra with a cheap burn, because the total delta-v necessary for that would be about as much as the second stage had to begin with (4000-5000 m/s). Thanks to the rocket equation, that means the second stage is not merely twice as large, but much larger, probably at least five times the size. And that in turn means the first stage needs to be that much larger.
In short, while propulsive reentry for a low-velocity suborbital stage is proven, propulsive reentry from orbit is known to not be a good idea.
Materials science has advanced quite a bit since the Shuttle was designed, and in particular, SpaceX has been experimenting with better, more robust heat shield designs for years. What's more, the Starship will have a much more favorable ballistic coefficient than the Shuttle did (since it will be much closer to empty), as well as a less demanding aerodynamic shape, making it easier to reenter with lower heat loads. So making the heat shielding reliably reusable without significant maintenance is likely to be quite achievable.