I don't think "trashy" was Anderson's intent with AVP at all. I seem to recall his intentions being that he wanted to honor the legacies of the original films, create a true prequel to Alien, etc...
The only film in the franchise that I'd say actually intentionally went for "trashy" is Alien: Resurrection. Jeunet's biggest interests while making the film centered around riffing on on the then-modern American blockbuster action movie style and structure.
Because he wanted to make a film that connected with American audiences - ie. do the job he was paid to do. Not sure how that translates to trashy or "trashy"?
Maybe "trashy" was the wrong choice of word but there was definitely a schlocky feel to what mainstream American action movies were doing at the time, and Alien: Resurrection
leans into that hard. A lot of people cite that as a problem with the film, since it is such a different style/tone than the previous Alien movies, but I see that as very much being by design. Jeunet wanted to make a "modern" American action movie first and foremost, an Alien movie second, and for me, personally, I find it very appealing in that regard. It oozes a vibe similar to what Dark Horse's Aliens comics were doing at the time, funny enough.
Whereas AVP set out with more of a direct intent to pay tribute to and respect earlier Alien and Predator films, and take itself more seriously... but ultimately it just kind felt bland and artificial to me.