Another way is, are the benefits of being a US citizen a fundamental right in the same way as Bill of Rights protections are?

It's the most fundamental right. By the time of the Civil War, the importation of new slaves had been prohibited for nearly sixty years. New slaves were created by denying to newborn children the right of citizenship. Given life expectancy at the time, nearly every slave that existed in 1865 would have been a slave because their parents weren't citizens. Or were illegals. Slavery existed because States and Congress had the power to say that babies born in the U.S. would be denied all of the protections of citizenship because of who their parents were. So that power was taken away from the government. All babies born in the U.S. that aren't immune from its jurisdiction (like ambassadors) - are automatically citizens. The government can't deprive them of that any more.

Just like we told the government that they can't imprison someone without a trial, we told them they can't take someone born in this country and decide they're not a citizen. It's the ultimate fundamental right.