Although I also can't help but admire the fact that Spielberg did make the centerpiece of his big budget family movie so incredibly unattractive and, well, alien.
I know I'm overanalyzing, but for me, part of the problem is that E.T. isn't truly alien at all. It's just a human shape put through severe distortion. It has all the human parts -- eyes, nose, mouth, ears, head, neck, shoulders, arms, belly, legs, hands, feet -- with only the proportions changed. Most "alien" designers in TV and movies just start with humans and modify them, and I've long found that tediously unimaginative.
I wasn't entirely clear on what Godzilla's character motivation was in this movie (I haven't seen the originals).
So, he wants to kill the MUTOs and he pretty much just ignores the humans. He doesn't care if they swim next to him, and he actually goes out of his way to avoid killing them.
Well, not entirely. He doesn't seem too concerned about the casualties from the flood he causes in Honolulu, or from tearing up the Golden Gate Bridge (note that only one of the many vehicles on the bridge was shown getting off beforehand). Ducking under the aircraft carrier may just have been about not wanting to run into a heavy and possibly painful object when it could be easily avoided.
And then when the MUTOs are dead he just gets up and shambles back into the ocean.
But he doesn't eat them or anything. He doesn't do it so sustenance. They weren't threatening his life or territory. He just sleeps in the ocean apparently. So why does he do it?
Well, Dr. Serizawa's belief was that he existed to preserve the Earth's balance by keeping kaiju like the MUTOs from propagating out of control. Which sort of reflects the spiritual component in a lot of the Japanese films, which sometimes portray Godzilla as a sort of protective spirit of Japan -- or, in
GMK, as a force of divine vengeance battling against the protective spirits Mothra, Baragon, and King Ghidorah. Either way, Godzilla is sometimes treated more as a cosmic force than an animal, and that seemed to be somewhat the case here. The scientific rationale was that he was the alpha predator whose appetites kept other species from overpopulating, but as you say, he didn't feed on the MUTOs here.
Then again, he was pretty badly injured and near death after the fight, so he didn't really have the opportunity to feed on them. Maybe he just retreated to the ocean to rest and recover. Takes less exertion to float in the water than to move on land, especially when you're that huge.