Making aspirin without acetic anhydride
I just looked in the literature, and it has been done even without catalyst in a "reactive distillation column" at high temperatures (patent US201113194873). If you can remove the water formed, you can technically drive the reaction forward, even if it is slow. A catalyst will just make that process easier, requiring less heat and time.

Phosphoric acid is a much weaker acid than sulfuric acid, and it does not have the same dehydrating properties. However, if you have a proper setup, it should do the trick. A big problem with phosphoric acid is that the most concentrated you can normally get is 85%, which means you still have to remove 15% of water somehow. If you don't, you'll likely hydrolyse any aspirin formed and get your starting material back.