My daughter was born on Christmas Day 2000 at 3:29pm. No denying, my sweet girl is a Christmas baby. It makes her special, but she does struggle a bit each year and I, as her mom, do as well. It's really hard to combine everything and make certain her birthday is a notable event. And, I make certain she always gets a separate birthday celebration and gifts. In our house, her birthday comes before Santa. I decorate the kitchen like a birthday party and we start with her birthday gifts. We also have a tradition that she wears some sort of cool birthday shirt on Christmas. The one this year was very tween-friendly and very cool!

We host our large extended family at our house that day for lunch and fun and we all stop promptly at 3:29 to sing to her and have cake and celebrate her.
My son was born December 7th, 1999 just one year prior to my daughter (12 months, 18 days apart). I do think their half birthdays are pretty cool milestones because they're both then closer to the next age. So, we definitely acknowledge them in this house and I generally get them a gift since theirs are both stuffed into the two and half weeks before and including Christmas. I also give them real gifts for Valentine's and Easter to balance for smaller birthday gifts or not smaller necessarily but more limited choice for gift since everything is so close. I love to surprise them mid year with something they love but didn't get at Christmas and birthday time.
That being said, I've given the option of having a half birthday celebration in lieu of actual (aside from family I mean), and they've never done it. My son's birthday parties, if he has one are anywhere from the 3rd-10th in recent years. We generally choose 17th-29th for my daughter. Back in the day of toddlerhood, I combined their parties around the 15th or 16th since it was halfway between. In 2012, my son chose an iPhone and family party in lieu of big friend thing, but my daughter had 6 girls to spend the night on December 29th and it was a success! So, holiday birthdays can be done.
All that being said, I'm not sure why people are so put off by the half birthday concept...my daughter generally took her birthday treats the same day as my son (which usually was his bday 12/7 or the day closest) because it made it easier on me and she got her day with her class too. But, if someone brought half birthday treats...why would it matter?