It isn’t about the birthday card. It is about someone with power over you demonstrating through actions that they don’t care about you. As an alternative think of this scenario: I prepared a presentation and arranged for the entire team to attend specifically for a manager above mine who had asked for it. He ghosted the meeting, and asked for a TLDR. Every single person in that meeting felt disrespected personally, and stopped taking the issue that we were meeting about seriously.

Note: If your workplace celebrates birthdays and you don’t want yours celebrated, just ask, they will almost certainly accommodate you.

I liked the system one of my old workplaces had.

If you want to celebrate your birthday with your team, you bring in your own cake. The manager will designate time for the team to eat it and maybe have some company beers at the end of the day.

If you don't bring in anything then there's just private words throughout the day.

The system eliminates any need to ask to split the cost or go through a budget request, since it's one dessert item once a year.