Not exactly, or not the only criteria, that's for sure. The words of the constitution and the entire body of US case law assigns 100% or birth abroad as aliens under the constitution, naturalized as Congress sees fit. The scope of that "Congress sees fit" is highly variable over the years. Nothing "natural" about it.
As far as amendment being necessary to change the constitution, that is demonstrably false. The government isn't bound by the constitution, unless you are a Pollyanna. The NBC clause is long dead. There is no need to amend it, just keep ignoring it.
The constitution IS meaningless - or rather, it means whatever the government gets away with and SCOTUS either ignores or endorses.
One can review the case law for academic curiosity purposes - or if money or some other thing of value hangs in the balance, to facilitate composing arguments. But the decisions are a figment of whatever 5 justices ate for breakfast. The US is a banana republic, in every sense.
A pessimistic view with a basis in truth.
I however am unwilling to permit this reality to continue with my consent. I choose to resist.