Well, I see it as a Separation of Powers thing.
Congress has the role you describe, but Congress is limited by language in the Constitution. The Constitution says that only a Natural Born Citizen can be president. Congress cannot change that. So the question becomes: What is the official definition of Natural Born Citizen.
The Supreme Court is often tasked with "explaining" the Constitution.
But perhaps we could have Congress make the determination.
My point is that I don't like "government" to just give a thumbs up or a thumbs down to individuals -- "This guy? Not eligible. That guy? Yeah, he's OK."
We need criteria. Congress can declare eligibility, but I'd like to see them define it: So-and-so is eligible, because they are a Natural Born Citizen, based on this set of criteria ..."
I think we currently have a situation where the Powers That Be just do what they want. The whole idea of a written Constitution is that government shouldn't operate on the principle that the Powers That Be just do what they want.
In another scene of this kabuki : McConnel has already said they will not touch the issue. That sends this’s hot potatoe to SCOTUS
Next scene. ( shame we can’t see who is pulling ropes off stage )
The constitution gives congress the power over naturalization. Therefore, NBC does not flow from the naturalization process.
NBC, like militia, are not defined as they were commonly used terms that everyone understood