Speaking personally, I'd say that none of the Godzilla movies I truly like have human drama I feel I have to wade through. A lot of the bad ones do, sure. Because they're bad movies, and only the special effects are at all worthwhile (with the exception of the few 70s movies made from about 75% recycled footage).
There are certainly many examples of fine human characters in the series. Not Academy Award material necessarily, but let's not forget that the first film in the series was a human drama - Godzilla was more of a plot device, albeit a memorable one. "Invasion of the Astro Monster" is also much more concerned with the humans than the monsters, and it's my favorite of the sequels (thanks in no small part to the contributions of the leads, Akira Takarada, Nick Adams and Akira Kubo). Even a lesser effort like "Terror of Mechagodzilla" managed to create a likable and tragic lead in Tomoko Ai, whose self sacrifice at the end is quite moving.
There have been plenty of examples of worthwhile human stories in the series. I don't see why an American film would necessarily have an advantage in that department. Mathew Broderick certainly didn't outdo any of the Japanese casts, with the possible exception of the cast of "Godzilla vs. Gigan".
