Someone will feel slighted no matter how well you treat them. I blame all of the 'micro-aggression' nonsense we've seen in the last couple of years and subreddits like /r/antiwork. This sounds like just the kind of thing that's encouraged there.
"The study found that when managers at a national retail chain failed to deliver birthday greetings on time, it resulted in a 50% increase in absenteeism and a reduction of more than two working hours per month. "
Most people I know don't care about their birthday (or being recognized) to this extent. Years ago, I worked at a place that decided to celebrate everyone's birthday that week with a cake and a 10/15 minute meeting (usually multiple birthdays were celebrated at the same time). As the company grew, we were having these meetings 3-times per week.
My 6 year old would care if their birthday didn't get recognized and similarly pout all day and seek this type of 'revenge'.
I wouldn't want to work with anyone that did this.