For clarity: the doc you cite says what I said (at least that's my interpretation) -- that non citizens have much the same constitutional rights as citizens. But they have to be living in the country (so-called US Persons). Hence it doesn't do you much good in the context of getting into the country. Source: I was a US Person for a while.
The legal argument is "yes they do", but the first few pages of the document are "but we understand why US citizens might assume the opposite because here's a laundry list of all the times we ignored that in the past".
As a non-citizen, the internal arguments of the US legal system aren't very interesting to me, but the historical record of how the US treats non-citizens is. The US has clearly demonstrated that in practice, non-citizens are not protected by the same rights.
