The main critique of trigger warnings are that they attempt to limit conversations to topics that people are comfortable with--however, that doesn't make a ton of sense because trigger warnings are essentially just giving a heads up when it isn't obvious that something triggering is about to be said. In other words, it says "X is about to happen and if you that will be too upsetting for you, then you should leave"
Bullying theaters into including another arbitrary warning label is irrational for two reasons:
1. MPAA regulations require movies to explicitly detail their blatantly troubling themes. Adding another layer onto this process is inherently irrelevant.
2. Things considered to be "triggering" are subjective. While it's fairly obvious that OP says "trigger" to refer to misogynistic/racially unjust themes, the exact definition of "trigger" is too vague. Things that "trigger" a third-wave feminist - such as the oppression of women - are worlds divergent to things that may "trigger" a domestic abuse survivor - such as any display of verbal disagreement.
As such, there's truly no significant warrant to implement a "trigger" warning system.
Movies have ratings to protect the public from adult material. One needs to use these ratings and common sense to decide if a given movie or play is something that they want to see. Using due diligence and reading about the movie or play should suffice in preventing exposure to shocking or mature material.
Trigger warnings are becoming increasingly common online: they are a shorthand label that indicates whether what you're about to see includes emotionally disturbing - or "triggering" - content. As helpful as these labels may be, it is going too far to slap these labels onto movies and stage plays. Audience members often know what they're getting into when they buy a ticket anyway, and oftentimes it has been shown that the label itself can be triggering.
Any decent review should be enough to let a person know what kind of content to expect in a movie or stage play. While intended to entertain, movies and stage plays also provide insightful commentary or emotional content. If it is necessary to provide a sound byte warning, rather than expecting the audience to check out reviews and trailers, every movie would need one. For example, the Titanic would have to have a trigger warning for tears.