I‘ve never gone public with an appeal; in my experience they reverse course quickly once you get a chance to speak to them most of the time. Sometimes it is like talking to a wall, but that’s certainly not universally true.
If your case is a simple misunderstanding, sure, it's possible that they will reverse course. They aren't impossible to deal with, and if it's clear they have made some obvious mistake they will generally quickly resolve the issue. However, the moment ambiguity comes up I have found that the people they send to talk to you don't have the necessary context or authority to really tell you what is going on behind the scenes, and the process moves forward so slowly that you risk hitting the deadline they give you. At that point you have few options but to go public, after which they will get back to you immediately with a fairly sane explanation of what the actual problem was or perhaps an apology if they were wrong.