I hate to use the argument but it's really sad that in 2018 we're still trying to make voluntary sex work illegal when every studies show that it 1) doesn't work and 2) makes everything worse for everyone involved. Also, the fact that the very president is known to have used prostitutes and that Washington politicians are well known to be big customers makes this hypocrisy really infuriating.
Backpage was taken down because they knowingly facilitated ads for unwilling sex work, sometimes involving children. You wouldn't have SESTA / FOSTA if you didn't have the pathologically terrible Backpage.
I've heard they started auto-filtering words (teen, loli, etc.) but that could be interpreted with different intents. These aren't strict liability crimes, so intent will come in to play and need to be proven.