Let’s start this by positively affirming that Ted Cruz is not the son of the Queen of England. So, that hypothetical aside, we can proceed.
First, with his father being a naturalized Canadian citizen, Ted Cruz even though he was a citizen of the USA by birth through his mother, also had Canadian citizenship. His parents were working in the oil industry in Canada at the time. They returned to the states when Cruz was 3. He was STILL a citizen of the US by birth.
Second, since Canada was discouraging dual citizenship at the time, and since he was only 3 years old, it is understandable that he did not consider any ties to Canada to be in effect.
Third, Cruz has always been a US citizen by birth. Finding out about a legal claim to Canadian citizenship, he renounced that citizenship simply to do the right thing.
All of that said, none of it matters toward Cruz having been a citizen by birth. He was. He remains so.
Despite having asked this hypothetical at least a half dozen times over the years, I never get a definitive yes-or-no answer.
If the answer is "no", the heir to the British throne, though having an American father, is not eligible, then it has great significance to this discussion, does it not?