This is probably one of the most consistently inaccurate criticisms I’ve ever seen of a popular TV show.
Scully did not remain skeptical past about the second or third act of the pilot. By the end of the first episode, she was pretty much firmly on board with the idea that there was clearly something weird going on in the world at large, and that didn’t really change much for the rest of the show. What would usually happen was, she and Mulder would get called into a strange case, Mulder would immediately leap to the most fantastical explanation imaginable, while Scully would counter with a mundane explanation. Then they’d investigate, and the actual evidence would usually invalidate both their original ideas and end up with some other weirdness. Scully was skeptical, in as much as the fact that it was aliens last week, doesn’t mean she automatically accepts it’s aliens this week, until there’s proof that it’s aliens. But she doesn’t spend the entire series running around saying things like, “Aliens don’t exist!” after being beamed up by the Mothership in the last episode.
And they actually made some effort into mixing this up by showing episodes where what appeared to be a supernatural/extraterrestrial event ended up having a mundane explanation after all. The “lake monster” that was eating people turned out to be an escaped crocodile. The “hungry ghosts” that were eating people’s internal organs turned out to be a cover for a black market organ harvesting operation. Even the “robot cockroaches” you mentioned in your OP turned out to be a case of mass hysteria caused by a few odd (but perfectly rational) deaths. Plus there’s a handfull of episodes where Scully is the first person to go for the supernatural explanation, while Mulder demures.
Part of the concept of the show is the idea of seeing monsters/aliens/etc. in our world, which is to some extent inherently limiting. If too much knowledge about this stuff gets out, it stops looking like "our"world pretty quickly, killing part of the shows premise. Personally, I’d really like to see an X-File/Buffy/Supernatural show where you actually see how growing knowledge of the supernatural alters the world, but X-Files wasn’t that show. C’est la vie, but I felt X-Files actually did a pretty good job of justifying that lack of change. For one thing, part of the overarching plot is that there’s a large and powerful cabal of people who are deliberately suppressing information about aliens, so it’s not just that people won’t listen, it’s that there’s people deliberately obfuscating the truth. And when the Conspiracy wasn’t part of a particular episode, they were reasonably clever about resolving things in a way that there’s no hard evidence left behind. And even then, they’d occasionally subvert it - there’s one episode where they find a Frankenstein’s Monster sort of guy, the result of genetic manipulation that created a hideously deformed, but intelligent and well-meaning person. That episode ended with Scully and Mulder convincing him to go public, and culminates with a hilarious montage of him becoming a minor celebrity, getting interviewed by Sally Jesse Raphael, and going out clubbing.
Of course, it was a long running show worked on by many different writers, and in some episodes the reset just don’t make sense. But that’s a problem with the medium as much as the show in particular.
For my money, where the show really fell down was in the “mythology” episodes, the (relatively, particularly for the time) continuity-heavy episodes that explained the over-arching conspiracy behind the show’s events. Famously, Chris Carter said he was making all that up as he went along, and it really, really shows. Nothing is ever resolved in these episodes, and what few explanations we got along the way were often contradicted by later episodes, leaving the whole thing a confusing, dispiriting mess. But the stand-alone, “monster of the week” episodes were often among the best things ever put on TV, even today, and are well worth watching.
