Not for your birth certificate, no. That is fully in the hands of wherever you were born and can’t change. Your driver’s license/ID depends on your place of residence but you still would have to transfer a legal license from one state to another while moving, which can require several documents to prove you are who you say you are and live where you say you live. But that’s really state specific!
Excellent points! There are definitely workarounds, but you have to look at what state options are available to you to make a plan that works for your situation. Some communities have trans advocates who can walk folks through that process for free, anyone looking to update their documents should see what resources are available.