I understand why the time is set to 10 AM, because of the wind on Route 13, but what is the reason the game needs to be set to February 14th?

maybe the game has a penchant for valentine's day that's related to happiness somehow? (i.e. get lucky egg) i don't know what else sadly...

Just checked bulbapedia and it isn't even for happiness

On feb 14th and march 14th you get 5 sweet hearts from the maid in Camphrier Town, which are used to raise Hawlucha's happiness

That explains the valentine's day (and the fact it works on march 14th as well in case valentine's and birthday are concurrent so as to avoid a birthday center cut scene), and indeed it's related to happiness albeit indirectly.