See, most self-identified feminists are against what is often referred to as "rape culture," which is to say the culture in which women are constantly being threatened with rape and normal sexual relations are warped so that men seeking women to rape is considered a norm. Rape culture is a culture in which men are encouraged to take sex from their partners regardless of consent and women are encouraged to be passive sex objects for these men. Women are valued for nothing beyond their ability to be suitable fucktoys for the guys in charge.
In a rape culture, if a woman should try to establish value on her own time by being her own human, say by going outside dressed however she wants by herself after dark and walking to a seedy bar where she plans to drink alone, she is threatened and "kept in line" with the threat of rape. In a rape culture, a man is treated as if he always has the right to sex whenever and however he wants it, with the implication being that if someone doesn't offer him what he wants right now he will just take it.
Sex work is an intrinsic part of this system. It is the device by which men are told that in fact, they do always have a right to sex and should never have to go without. It is the palliative that men are offered in the hopes that their big bad virile selves don't go rape us defenseless girls. Men are offered this palliative because it is unthinkable that they actually be taught not to be sadistic and woman-hating in the expression of their sexuality.
Not only does sex work support the "satisfy us or we will violate you" aspect of patriarchal rape culture, it confirms the world view of the men who believe that this is the right way to live. Women are objects which you can buy the same way you buy tupperware. Women are fucktoys- you can even tell them so and they will agree (because you paid them too). Women always want it (because you pay them to pretend they do). It's impossible to rape something that is less than fully human and that always wants to have sex with you anyways. This is what men are paying to be told when they see an escort, stripper, or any other woman who sells them the illusion of sex.
They are paying to pretend that a woman cannot say no. And they are still imagining that no woman can refuse them when they get into their cars and track down some coeds to oggle. They are paying women to confirm and conform to patriarchy and rape culture. The elimination of the sex industry would eliminate one of the ways in which rape culture perpetuates itself. The elimination of the sex industry would and should be a feminist-approved route to a world where women are treated as more than merchandise and men understand that it is not the rest of the world's problem if they can't find anybody to engage in consensual sex with them.