The doctrine of sovereign immunity is grounded in the divine right of kings. If the right to rule is conveyed directly to the king by God, it stands to reason that the king is subject to judgment only by God, and is not subject to the law. Hence, the king can do no wrong, at least from a legal perspective.
The argument that Trump cannot be guilty of obstruction of justice because he is the chief law enforcement official of the US sounds more than a little like this line of reasoning. You can see why it would appeal to him. Unfortunately for him, our political system is based on contract theory, not the divine right of kings. In our country, the people are sovereign, and they have chosen to divide power among the various branches of the federal government and between the federal government and the states. There is no such thing as the chief law enforcement official in our country. As a result, the argument fails miserably.