You are correct.
But can a dual citizen get the security clearance for that position? I don’t think the DoD would allow it. They don’t even allow passports holders of other nations to have that clearance of lower level clearance.
http://www.dod.mil/dodgc/doha/isp.html
The President is not actually given a security clearance. As the head of the Executive Branch, and by virtue of winning the election, he is entitled to access all classified documents and information. He's the boss, essentially, so there is simply no way to deny him access to any secrets he wants to see.