While getting hacked and "private" data being leaked, private messages among them, I believe the more important issue with responsibility of a hoster would be, that they would have to remove hate speech, harrassment and such. They can't just have you sign a waver and be done with it. Maybe discord can get away with it. But on a regular website, if I harrass you with pm, there has to be means of the site removing the messages and banning the account. The harrassed account being able to "block" the offender is just not enough.
This just all translates to the high maintenance of such features.
So if you want to have a different wording: it would cost money to have such a feature and people would not pay for it.