That shouldn't be an issue. Most are issued a few days at least after birth. It's not less official because it's old. In fact, the only "original" birth certificate is the one issued shortly after your birth. Others are certified copies or certified reissued ones. You shouldn't need that for a passport, whether you're 10, 21, or 50.

Here are the requirements from the State Department website:

https://travel.state.gov/content/tra...-evidence.html

It appears what you have meets them all, including issuance within one year of birth.

The point of a birth certificate is to establish citizenship and verify birthdate, which in your case it establishes because you were born in the U.S.