Mechanisms are explanative, and not predictive.
The first step of the reaction mechanism is protonation of the anhydride because that is the first step of the reaction. That just is. If the anhydride is not protonated, step 2 simply won't happen.
This does not mean that the salicyclic acid is not protonated. It just means that the protonation of salicyclic acid is not necessary to explain how acetylsalicyclate forms. Proton exchanges are at equilibrium. It is likely that the salycilic acid is protonated during this process. However, that protonated salicyclic acid may then protonate acetic anhydride and proceed as indicated in your link. For the purposes of describing the mechanism, it is not necessary to distinguish where the proton that protonates the acetic anhydride comes from.
So your question really is self-answered, where you say, roughly, 'Such a protonation does not lead to product directly'.