Seeing people wish you happy birthday on there by clicking a button encapsulates how false the narrative is; if the reason you remembered my birthday is that Facebook told you then we don't have a relationship where you should be sending me a birthday message.
It's only as genuine or fake as you make it be. I'm sure you don't keep in touch with 100% of the people who made positive contributions to your life. You can take this as a less-awkward opportunity to wish them well, and maybe get back in touch if you feel like it, instead of living life as if their existence doesn't matter to you. Or you could not... you could do something else, or you could do nothing if you don't want to do anything with people you don't regularly keep in touch with. It's up to you how real it is and what it means; that you can only think of false narratives for it doesn't mean it is or needs to be that way.
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It's only as genuine or fake as you make it be. I'm sure you don't keep in touch with 100% of the people who made positive contributions to your life. You can take this as a less-awkward opportunity to wish them well, and maybe get back in touch if you feel like it, instead of living life as if their existence doesn't matter to you. Or you could not... you could do something else, or you could do nothing if you don't want to do anything with people you don't regularly keep in touch with. It's up to you how real it is and what it means; that you can only think of false narratives for it doesn't mean it is or needs to be that way.