My gut tells me that the big red button might not even get noticed.
But I can imagine that people accessing information about domestic abuse might not necessarily have regular access to internet connected devices, they might not know the best ways to act under stress. Maybe they won't notice the big red button, but maybe there is some chance that they will notice it, and therefore some chance it will be beneficial to them in that moment.
1. Lots - if not most - traffic is from mobile these days. Most people already know the fastest way to exit a page on mobile - the home button/action. Adding anything else is just adding confusion. 2. Unless you are going to great lengths - ie pre loading a page and maybe dropping parts of the dom and dealing with evidence in the history - are you actually doing anything much to help the user exit your site? How motivated/skilled a person are you defending against? 3. If your exit button is just a glorified link or redirect what is the point? It will still be in the history and if they have slow internet they could end up just staring at your site while the redirect loads. 4. For some organisations having such buttons is more about "showing" they have it than how useful it actually is to the user. 5. I have tried to push for a page/link to basic internet safety information. Educating visitors would be much better than trying to engineer their personal security day. 6. I've struggled to find good academic/research work on such features. Seems like it would be a good area for a UX researcher but I've not found much actual work.
Other methods for leaving the site still work. Even if the button isn't the best way to leave the site, if it helps in more cases than it hurts then it's a net benefit.
These buttons are essentially panic buttons, and when a person is panicking the big red exit button might end up being the only exit they can find.