I don't think anyone believes that the child is born a Muslim in a fiqhi sense, that if he/she dies, janaza is prayed over him, etc...
What people mean is that a child is born on his fitra, recognising Allah as the only Lord and the only one deserving of worship. Meaning, he does not have to contemplate deeply on the emergence of the creation to prove the existence of a creator. He simply believes in a deity from birth. This does not necessarily mean that he is a Muslim in fiqhi sense.
So whilst it is good to clarify that a child is not born a Muslim in fiqhi sense, it is also important to clarify that the child is actually born on tawheed, such that he does not need to reason the way the mutakallimun want him to, to discern God's existence.