That's part of it, maybe most of it. Another part is to preclude any other country from asserting a claim. The War of 1812 was fought in part over British claims of power over US citizens, who were, according to the British, British citizens because they had British blood in them. As British citizens, they had a duty to fight war for the British, and many were compelled to do so.
That sort of risk could be messy if the US president was part British, e.g., by being born in Britain (but made a US citizen by e.g., Naturalization Act of 1790, or 1795, or 1802, etc.)
Even Obama is more American the Ted Cruz. Sad to Say.