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It's almost never broken. I have only been turned away for a broken machine one time ever, and McFlurries are more or less a monthly treat.
Sorry to break the internet meme, but I'm pretty sure this isn't all that common.
Fortunately there is objective data: https://mcbroken.com/
It's probably locational and possibly the time of day. Since I understand cleaning them are a pain, certain locations may not prioritize it properly, or the downtime is always when you would go. The one by me for example has been down all of the 10+ times I've asked for something, but I know family members have gotten some ice cream there, so it's not like permanently busted or anything.
The documentary on it explained that the cycles are timed to ensure proper cooling, and not warming too much for too long for sanitary reasons, so if an employee over filled the milk by a small amount the machine would sense the extra process time and default to needing a dump and clean. There was that raspberry pi device that would bypass that but for obvious reasons mcdonalds repair company lobbied against it.