Your second paragraph, as to whether "any Muslim country, sect or person can be trusted—anymore than communists or Nazis" --- is not a valid analogy. The valid analogy would be between people who are born Muslims, and people who are born German or Russian. But why?
It's because Muslim law regards every human being as coming into this world as 'naturally' Muslim. Moreover, every child of Muslim parents is a member of the Umma, whether he or she has ever professed or practiced the religion, or even knows anything about it.
In other words, a person can indeed be "born Muslim" without implying personal allegiance to Islam, just as a person could be "born German" or "born Russian" without any personal allegiance to German National Socialism or Soviet Communism.
I would want to put the "bright red line" not at "Were you born Muslim?" but "Do you advocate Shari'a?"
I don't think a personal who advocates shari'a should e allowed on our shores under any circumstrances, not even as a member of a diplomatic delegation from an Islamic country.
Neither do I propose that Muslims per se should be eligible as refugees under pretext of being religious exiles. Alawites, Sufis, Druids and Baha'is, maybe. Christians, yes. Yazidis, yes. Sunni or Shiite Muslims, no.
Your born Muslim conflates nature and nurture. Under your definition, “born Muslim” is equal to raised Muslim. However, even Muslims do not believe that as their constant fratricide proves. If one is a member of the Umma, one is the enemy of everyone else, unless they truly recant their faith(or reject their birthright in your scenario).
I would agree with your Shari’a statement, but that brings us full-circle to the fact that anyone who advocates Shari’a also advocates lying for the cause. There is no way to ascertain that one is telling the truth.
As for the immigration, I would have no problem banning anyone who cannot swear allegiance to the country. I would also have no problem with deporting American natives with the same America-hating sentiments.