I’m glad it’s good enough for you but it’s not good enough for many. Oh and by the way, you can get into the military with NO birth certificate and that’s a fact. Check it out.
As a 20 year Marine Corps veteran, I don't need your advise on service eligibility requirements for the US Armed Forces. Yeah, you can enlist. But, no. You can't be an officer and no, you can't handle classified materials, and no you can't fly our planes unless you're a US citizen. And, if you're not a US citizen, you still need a BC from your country of origin.
You don't have to be a citizen to join the military, although you'll need documentation of legal resident status. You do have to be a citizen to be a commissioned officer, but for that naturalization paper, or other proof of citizenship, will serve as well as a birth certificate. You'll also need some proof of age, since there are age limits for both enlistment and commissioning, although they've recently been raised to pretty old values.