> It's extremely unintuitive that classical systems should minimize anything like a time or an action.

If I get to define the measure arbitrarily, then I can always find a measure that something else is always a minimum of. So in that sense it's not surprising at all. The interesting question to me is why should that seemingly arbitrary measure be action? What does that even mean, physically? I have no intuition for it.

> Think about it: they travel along the minimum path,

I'm already stuck here. What would it even mean for a particle to have an "action" (whatever that is) that is not minimized? Like what would that look like, physically? I understand what it means for distance not to be minimized, but action isn't distance...