This varies by state for birth certificates.
Learn how to get a certified copy of your U.S. birth certificate. The process depends on whether you were born inside or outside the United States.
www.usa.gov
The cost and procedure varies.
But, you'd need one to get a passport, join the military, get a US government job, and for many other things. So, it isn't just a voter registration issue here. You need one to get a Real ID for example. Without one of those, you are paying $45 now to get through TSA screening at an airport to fly domestically.
So, on you point about the cost of a birth certificate, it's going to be necessary for travel, jobs, and other uses today. Your singular voter registration argument fails.
As for a passport, you don't need to get one if you don't have one. So, that's a moot point.
You should already have a social security card. It's rare for anyone not to have one today. Proof of residency is as easy as utility bills or other mail showing your address. A passport is unnecessary for any of that.
Illegals have been given Real ID's. Commercial driver's licenses are one example of that. In California, and many other blue states, when you get a driver's license--which California gives to illegals both a personal version or a CDL until recently--you would get registered to vote at the same time. California's system registers you automatically unless you choose to opt out. That means a sloppy or uncaring DMV clerk could be issuing lots of illegals voter registration just because it's easier than not doing it.
Actually, it's you that's the ignorant ass here.