The President should be nuanced enough to be aware that calling the press an "Enemey of the people" is something tin pot dictators do on the eve of their ascension to power. The phrase is deliberately inflammatory.
If this were all just an honest mistake he wouldn't have repeated it over and over.
He said clearly that he wasn't talking about the entire press, just the people who lie to to the public. If he's the liar, he's ironically talking about himself. If the mainstream media is intentionally lying to deceive people for political purposes, then he would be negligent not to speak against it.
So I'm not disagreeing with you. But this really comes down to the question of who is lying.
If this were all just an honest mistake he wouldn't have repeated it over and over.