Well as much as I want a kaiju driven flick, but the majority of movie go-ers want to see humans getting in on the action rather than leaving the world's fate up to a kaiju radioactive dinosauromorph, a volcanic pterosaur, a majestic divine moth, and a guardian ape.
However the majority Godzilla franchise have always been human driven, examples:
Gojira (1954): The whole plot centered around a paleontologist trying to identify Godzilla, his daughter and her boyfriend a JSDF officer, and Dr. Daisuke Serizawa being convince by the JSDF officer to use the Oxygen destroyer against Godzilla to save japan. Special mentioning go to the average Japanese citizens dealing with godzilla's rampage.
Mothra (1960): Centers around several journalists and a greedy CEO who travel to infant island to investigate the strange anomaly of a thriving forest on a other wise heavily radioactive island. Corpoartion suits exploits the island by kidnapping the shobijins fairies, forcing them to be entertainers. In their distress, they summon Mothra through a song to rescue them. Which the journalists must find a way to rescue the shobijins before Mothra destroys Tokyo and Rosuland.
Most of the Showa is kaiju driven with a few notable examples like Terror of Mechagodzilla.
From what I can remember the Heisei series of Godzilla was very human orientated with them developing Mechagodzilla, having a character that can communicate to kaiju, and even time traveling humans that created both King Ghidorah and Mecha-King Ghidorah.
Godzilla (1998): Centers around a radiologist, his journalist ex-girlfriend, a camera man, some french guys, and briefly the mayor of manhattan. Team up to stop Godzilla.
I need to watch the Millennium series, so no examples here.
Godzilla (2014): Centers around a young military man trying to get back to his wife and son while trying to survive through a kaiju fight.
Shin-Godzilla (2016): Haven't watched it yet but from what I hear its mainly about Japanese bureaucracy while in amidst of a kaiju attack
Godzilla: King of the Monsters (2019): Eco-terrorists and a turn-coat Monarch agent free king Ghidorah from his icy prison. Humans cripple Godzilla, allowing Ghidorah to become Earth's new Alpha. Humans having to resuscitate Godzilla via a nuclear bomb to save earth from being xenoformed by King Ghidorah. Humans distracts King Ghidorah so Godzilla and Mothra could have a fighting chance.