Very good.
This is precisely the point. We must not treat birthdays as “innovation” in Islam.
Birthdays have nothing to do with worshipping Allah. They are just a gathering of people to enjoy their life.
pretty much like the Mehndi celebration before wedding.
A middle eastern may not even understand why a Pakistani family is enjoying Mehndi. And in his ignorance may think, that Pakistanis worship Allah during Mehndi celebration, and hence it is an innovation a bidaah.
As you said, if something is dealing with worshipping Allah, then we must check if it is new or old. Bidaah or not.
But if something is totally in public sphere and nothing to do with worship, then we should not take our sword out to slay it.
Thank you.