I think he had to make a public showing of some sort, seeing that his birth certificate is Canadian. Some people would interpret that as "significant," somehow.
The Cruz camp finds the same thing you do, that he was never a Canadian citizen, and Canada had no claim on him.
But Cruz's office says that so far as it knows, the senator has no Canadian citizenship.Ted Cruz releases birth certificate - WaPo, August 19, 2013
I've said all along that there are questions regarding Ted Cruz in this regard, but that these questions could potentially be resolved in his favor. He's renounced any claim that Canada may have had. Canada needs to renounce any claim also. Cuba, I believe, lost any legitimate, legally recognized claim upon the father of Ted Cruz due to the Revolution there and his becoming an exile. He threw off that allegiance, which in turn removed any claim upon his son.
The whole matter is simultaneously more complicated and less complicated than people have been making it out to be. It's not a black and white, clearcut matter other than in the instance of people born in this country to parents who are it's citizens. Others can be natural born citizens, but there are questions to resolve, the resolution of which can be favorable, or unfavorable. This ties directly back to Chief Justice Waite in Minor v Happersett, among other pertinent cases.