I can't imagine why anyone would have a problem with this. It's not like you're going around town to different restaurants every day for 2 weeks asking for free birthday things at each of them! Some restaurants don't even give you anything free, they just sing to you. As far as which restaurants do special things for birthday's, I think most of the chains would provide a piece of cake or something similar. I remember seeing this at Applebies and I think Outback does it too. Damons by us brings a dessert and puts your picture on the TV screen in the clubhouse. Embarrassed my kids something awful!
When I waited tables at Farrell's, birthday's were big deals. We did give the birthday person a free sundae, and we sang to them and made big announcements. It wasn't at all uncommon for a family to come in and celebrate several birthdays that were close together. Basically, they'd just make one trip for all the birthdays that month.
Go for it.