Alright, first things first, light doesn't choose the path of shortest time. When you say "chose," you mean a conscious entity picking a certain course out of numerous courses. That, however, is not the case.
Fermat's principle can be derived from Huygens Principle, or perhaps is a consequence thereof. Now, to answer your question: light propagates (/traverses) the path of shortest time solely because that is how we can describe it mathematically.
Simply put, since the best way to describe diffraction (and/or light propagation) is by Huygens Principle, its mathematical consequences (one of it being Fermat's Principle) must be accepted as well. That is, we must accept that light "choses" the path of shortest time, by definition, so that we can have a consistent system of optics. That said, if it turns out by the aid of advance instruments, that this description of light is anomalistic, then we will obviously abandon Fermat's Principle.
For the time being, however, it suffices to say light takes that path of shortest distance solely because that is how we have defined it.
- Huygen Principle is the best explanation.
- Fermat's Principle is a consequence of (1)
- therefore, in order to have a consistent system we much accept (2) as long as we do not face physical anomalies because of it.