"Even if she is a girl, she could be great help to us." This line from the English dub in Godzilla vs Biollante. As far as I can tell, in the original Japanese language, her (Miki) sex has nothing to do with the military writing her off and everything to do with most of them not believing she has ESP till after she makes Godzilla change course.
The next one, I'm not sure if it counts as a Godzilla cringe thing or a dumb misconceptions non-fans have thing but...real life military officials saying they have the weapons to beat Godzilla if he was real.
If Godzilla was real we'd be skreeonked. We be skreeonked. Like our fudge is gonna be packed, we are getting skreeonked so hard in the ass. Unless we found a way to build super weapons like masers cannons or Super-X or Mechagodzilla to drive him away we can not do jack poop to him if we was even remotely like his movie counterpart because Godzilla, by his very nature, defies the laws of physics and nature as we humans understand them so far. In the words of 2 of Godzilla's "fathers".
"The theme of the film, from the beginning, was the terror of the bomb.
Mankind had created the bomb, and now nature was going to take revenge on mankind." - Producer Tomoyuki Tanaka
"If Godzilla had been a dinosaur or some other animal, he would have been killed by just one cannonball. But if he were equal to an atomic bomb, we wouldn't know what to do. So, I took the characteristics of an atomic bomb and applied them to Godzilla.
" - Director Ishirō Honda
And the crazy thing this was said not once but twice, once by a U.S. military official and another by a Japanese military official(!) Even in films where Godzilla loses to the JSDF, most times the best they can do without Mechs is trap him in ice, a volcano or make him sick.