Not really. I'd hope that there were more important things in most people's lives than what nation they live in or were born in. Even then, I know a LOT of people that are Texans before they are Americans......and when I was in the Army I met a TON of people from the deep south that considered themselves southerners before they considered themselves Americans so even if your place of birth or place of residence is the most important thing to you, it still might not be at the national level. I think it would be important for the president to not have loyalty to any other nation over the US, but I don't think that has to be the highest priority in his life so long as he abides by the constitution and doesn't value any other nation over the US or do things intentionally harmful to the US.
You're actually very wrong about that. He's a true believer. His core political/religious mantra has not changed (I'm sure it's grown/devolved/modified) in 20ish years.
You are totally not the demographic that Cruz is aiming at with that comment, or any other. I think it will resonate very well with evangelicals and, ironically, with patriots. What is the Marines' motto? "For God, Corps, and Country"? Anyway, I think Cruz is atop his own priority list. If it was God or country first, or even conservative or Republican, it'd be an improvement.
...Biggest reason for that disconnect between many "Christians" and their "values", in my opinion, Dubious... ...is the falsehood of material gain following devout piety and adherence to the "Christian" doctrine. ...which, as you pointed out...is WAY off from what Jesus of Nazareth (the "author and finisher") of "Christian" faith outlined. Look at who is out front on the "Christian" message...and look at what their surrounding trappings are...and it's really not hard to see why so many people accept an inherently contradictory message... Following the money has never been easier to do than in religious circles. Not to say that there's anything, in particular, wrong with that, from an "American" perspective. Separation of church and state facilitates that dynamic like no other force in human history before it. Snake-oil and used-car salesmen have been around as long as robber-baron oligarchs here. American "worship" of the dollar has done as much to "elevate" a "Christian" nation as any sermon in anybody's pulpit. And also, like you said, Dubious...if you're really interested in what somebody "believes"... ...just take a look at what they're doing when nobody's looking... ...(which, by the way, is a lot harder to do nowadays, what with all the pesky camera phones and armchair inexperienced filmographers messing up everything)....