The example you are giving on the other hand does not. Why would the outcome of an attempt to persuade someone be random? That's not comprehensible and just leads to a quickload fest. Of course, there should be a chance of failure. But in most game situations that should be determined by a skill or the lack thereof - or even better, by your actions, e.g. the dialogue choices you make in your attempt, in conjuction with the skills of your character.
The thing here is: an element of randomness doesn't mean each outcome has equal probability. An element of randomness could be, say, 5% chance of failure. This simply helps by adding some non-determinism, because even the most skilled [anything] can fail at what they do. What I'm saying is: transform this into a game element. So instead of having one fixed dialogue option that tips the balance to either side, have a slower progression where, for instance, a persuasion attempt failure doesn't mean you cannot keep trying. It just means you have to try harder. Eventually your poor choices might close this opportunity for good.
What I would like to avoid is that other idea of situations being "determined" by either skills or dice rolls. For instance, a character with low persuasion trying to convince someone to do something, and failing, but due to that character's knowledge of certain events slowly tipping the balance in his favor. The alternative would be "if persuasion >= threshold then pass; otherwise fail." This means slowly approaching that threshold by other means. In either case, I believe the chance of failure should always be present. Think of an experience diplomat giving a speech and accidentally burping or throwing up in public, this should fail to convince at least one person (if anything because he couldn't even deliver his idea clearly).
I think the problem with people blaming the dice rolls (or RNG) is a game's poor (or complete lack of) mechanisms of explanation, specially in the case of failure. I think saying the "RNG arbitrarily made you fail" is like saying "life arbitrarily made you fail".