Regarding the "registration"of the aircraft then there is another situation that I have not seen commented on.
US ownership of anything is considered property tax which varies from State to State and is usually an annual tax and can be quite expensive.
There are a few states that do not have property tax, Delaware and Nevada used be the two that I was familiar with.
You will find that the registered US address for companies that provide this registration "facility" are registered in one of these states.