Actually, thinking about the vocal melody, I can picture Hetfield singing it. I can very much see how that song sounds like Metallica.
The vocal melody is so far from Metallica, I don't even know where to start. It has a range of an octave rather than grunting out the root note, and it's not at all a scale Hetfield has ever sung in. And whatever key you put the song in, it would be way out of Hetfield's range anyway. If you can picture Hetfield singing that, you have an amazing imagination, my friend.
The verse melody (or lack thereof?) in Constant Motion is pure Metallica though. It's mostly the E root note, with the occasional third and lower 7th. That is the limited kind of melody Hetfield is known for, not a wide spanning Mixolydian melody in a high vocal register.