Birth certificates are sort of anachronistic documents. Public schools, for example, require them in order to register students, but why? Why do you need a piece of paper to prove you were born? Isn't your mere existence enough? And it's not a citizenship thing, because (in Massachusetts at least) we do not require proof of citizenship or legal residency to attend public school, just a birth certificate and some proof that the child lives in the school district (like a utility bill, etc.). And it's not like they have pictures on them, so they're not really a very effective means of identification.
This is probably another case of "this is the way we've always done it," and no one really gives any thought to whether it's necessary any more.
Birth certificates here haven't historically carried as much weight, and you can still find people born before the '70s who don't have them at all. They're actually gaining in importance now, since they're a critical first step in proving one's identity to get other forms of ID, such as a driver's license, which will now be compulsory to vote (not-so-fun fact: I still don't have enough ID to get one of them, which means I definitely won't be able to vote in the next election).
Historically, you'd only really've needed one to apply for a social insurance number (akin to your social security number, but with a more ominous acronym), and this being the backwoods of Alberta, it's not like finding employment without one was ever a big deal, especially since a lot of people grew up to take over a family farm. That was the only reason my grandmother ever got her birth certificate: at the age of 18, by which point her exact date of birth had been lost to the mists of time, with the lovely phrase, 'About 21 April,' printed on her birth certificate. My grandfather's birth certificate, if any existed, was left in Poland when his family was deported by the Soviets in the Second World War, and he managed to live his entire life without either a birth certificate or a citizenship certificate, only a Displaced Person's Identity Card.
All in all, I agree quite fervently with your sentiment: I exist, whether or not I have a piece of paper that says I do.
I find the whole ID-centric culture we've built to be kind of sad, and while I get the concerns with security fraud prevention, it doesn't make living without an ID any easier. If having a photo ID is going to be mandatory to do such basic things as vote, I really feel at that point it's incumbent upon the government to provide one for free: anything else amounts to a poll tax.