How does that make it simple? If they want their day in court, they will get one. Lots of people use false identity. Go to some liquor store and ask the clerk how many lily-white kids try to use a fake id to buy booze.
If they lawyer up, they'll get in the system and will stay there for quite some time. Maybe they have kids born in US, maybe they can prove some familial ancestry, they need a continuance, etc. It is not as simple as you think.
No, unfortunately, following the law is never simple for some folks.
In this case it should be. All they have to do is produce a valid verifiable birth certificate. Same as your example of the underage kid at the liquor store.
Having kids born in the US while the parent is an illegal does not make them immune from deportation.
If you claim you were born in the US, you'd have to have a very good reason why you can't lay your hands on that type of documentation.