I think that would work, and the code for that part doesn't actually sound too hard. (Of course, my only experience is in Turing and Python). I like the idea where you choose when you were born in the season, although it sounds a little harder. But mechanics aside, you know what would be cool? If, on your birthday, the mayor and a group of NPCs who are X number of hearts with you at least, will organize a party for you in the community center if you've repaired it, or in the village square if not. They will sing "happy birthday", (which is finally public domain as of march 2016), and maybe have cake be present, but not edible necessarily. It could be set up like a festival, where everyone has 1 thing to say about it, and then you talk to the mayor and initiate a cut scene. In the cut scene, everyone could give you a present, which might be something cool like a keg or furniture, or it could be something lame like parsnip seeds in late spring.
I've had the problem where the birthday menu blocks cut scenes. Otherwise haven't found any problems.