> It’s not a proper noun

The President, within this context, identifies a single entity. As such, it is a proper noun.

Analogy: there are many continents. But if we're discussing Brexit, the Continent is a proper noun. I don't think it's incorrect to not capitalise. But it's certainly gramatically okay, and not in the same bucket as The Nutters who capitalise Random words it Looks like Legalese.