Okay, here is a completely different idea for birthday. Make it a reward for getting to full hearts with every giftable villager.
After getting to full hearts, next time you go into the community center, or maybe the bar, on a normal day, there is a cutscene. It's a surprise birthday party! Afterwards, that day is now your birthday, and you get happy birthday messages from everyone on that day, and there will be a letter in the mail from your parent. No rewards, no perks, no gifts. Just a fun little surprise cutscene for maxing out on the hearts, a yearly letter from parent, and the addition of your birthday to the calendar. I think that is plenty. Maybe also a special birthday meal if you talk to your wife on that day?
I'd be happy with just the cutscene. It would make a good capstone for reaching full hearts, and there really isn't anything like that in the game so far.
I don't really want more gifts, as was mentioned, I already get plenty of gifts. A fun cutscene as a reward would be nice though, and it it's a fun way to finally mark the day in your calendar for older players.