I think everyone should have rights but him being American is significant because most people's impression is that as an American the right to a lawyer is protected by the Bill of Rights to our Constitution and Americans take any violation of the Bill of Rights very very seriously.
Those protect people not citizens. You’re right that’s not how it’s popularly understood but it’s why we have Gitmo: it’s an attempt to apply a thin veneer of legitimacy by pretending it’s a military proceeding (as if theres exceptions for those in the documents you mentioned) and keeping it as far outside the purview of judicial legal analysis as possible.
People dispute what the government “should” do, but when it comes to citizens, there are certain things the government “must” do, regardless of what people think.
Because the purpose of a government is to keep its citizens safe, not act as world police.
Do you really believe that giving lawyers to people you have already detained is acting as "world police"?
The detainee is a citizen and as such is afforded the same protections as any other citizen. By definition, not providing him with an attorney is not keeping him safe.