:: British common law recognized that children born outside of the British Empire remained subjects ::
^^provided BOTH parents were British subjects at the time of birth^^; it takes 2 to tango.
The US, on the other hand can point to narrowing that simple requirement to being in service to the USA (military or State business)
I think you miss the point.
The critical fact here is not the narrow or broad specifics of what constitutes a citizen at birth but, rather, that a citizen at birth is indeed considered a natural born citizen either by jus sanguinis or jus soli.
No one is claiming (that I’m aware) that Cruz wasn’t a citizen at birth via jus soli. The only debate, weak as may be, is that a citizen at birth is not neccessarily considered “natural born”.