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It kind of depends on what era we're talking about. In heisei a vast majority of the monsters have ties to Godzilla so if he were never to exist the only monsters that I think would show up would be Rodan, Mothra, and Battra, all of which are at least a tier lower than the big ones that also show up in that series, leading me to believe we'd be fine.

Showa era would kind of be screwed, and Ghidorah likely would have destroyed the world if only Rodan and larva mothra showed up to stop him.

Millennium, depends what movie again.

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The very first scenario you highlighted actually happened in my own main written series, Colossus By Design.

Giant monsters showed up in the not-too-distant future, forcing Humanity to unite and create giant robots to combat them. After years of battle, the giant monsters weren't seen again for a long time, and Humanity engaged in WWIII with their giant robots, causing so much destruction that the world governments decided to stow away these machines indefinitely.

Of course, the giant monsters resurfaced later on, and so did Godzilla, whom was only tangentially known in inner circles and infrequently appeared throughout history.