People always talk about how "bad" the human stories in a lot of Godzilla films are, and while there's some truth to that, the best ones have always been nice and simple.
In the case of most of the Heisei era films, you had science squad vs Godzilla vs third party most of the time, and some limited drama between science squad characters. By and large the same for the Millenium films.
In the Showa era, it was always a mix of genres. In Godzilla vs Mothra's case, a crime drama mixed with kaiju hijinks. Or an alien invasion story.
I wouldn't argue any of these were brilliant, but they were entertaining in their own right. It's why those older films worked even when there was ultimately only 20 minutes of actual monster action across 100 minutes of runtime.
You can absolutely make a fun and entertaining human narrative within a Godzilla film, it's just that King of the Monsters especially failed at this not because it was doomed to fail, but because the actual human story sucked ass.
If you cut the poorly written family drama about a genocidal mother and her issues out of the movie, you'd have a perfectly functional tale about Team Science vs The End of the World.
Hell, keep the environment terrorist dudes because that falls in line with Goji stories.
But don't even try to get me to care about this family and the fact that the mom is a total monster. Do not attempt to get me to feel "sad" by the end for her plight.
I think Skull Island is the PERFECT example of a Godzilla-style human story that's plenty entertaining and light.