I think it would be weird to not choose, as day 1 doesn't need the day for the farmer to have his/her birthday, and picking a birthday makes the existing roleplay aspect even bigger, so it'd only be logical to choose.
I use the mod (when playing again), and have all characters a different birthday, especially the 2 based on myself and my father.
The only thing i'll avoid is using an event day, as that makes me unable to talk to some NPC's.
There's been a mod for a long time now, so i think it'll be easier to think how to make it, if it comes to the original game.
My father's birthday was on May 24th, but due to the Flower Dance event's date, i set it to 23rd of Spring for the character based on him. Close enough.
I just don't see why we'd have it chosen randomly(unless using a randomizer) in a game that's based on the roleplay aspect more than anything else.