If you're sitting next to the person or on voice chat to tell them not to click next in the conversation, why can't you just tell them to press "l" to follow the conversation?
I agree this system should be improved, but there is absolutely no reason anybody should be unable to follow the conversation as is.
As mentioned by someone else before, reading conversations in a log kills the immension. I'm standing next to my partner, why can't I follow her conversation directly instead of looking it up from some kind of book?
And then there's the issue that the dialogue log restarts with every new conversation.