I think the analogy best captures "some things are easy, some things are hard". For instance, if I were the stakeholder and this analogy was used to convince me that moving a web app that uses a database is hard, I would still wonder what you really mean (and I have lots of experience with Kubernetes). Probably better just to say what is actually hard instead of reaching for an analogy unless you just want to get past the stakeholder's "BS firewall" easily so you can get back to work. If that is your goal, absolutely use any persuasion tactic at your disposal. However, this scenario has a massive bug: the non-technical stakeholder didn't delegate decision making to someone that they can trust.

The analogies shouldn't be needed at work for sure, but they work well for people who aren't industry and don't especially care. Eyes glaze over at anything beyond "database" but people know about moving toilets so they appreciate the analogy when you're at a family reunion or something.

Perhaps, but those people are the bug in this hypothetical. They should instead delegate decisions to someone who is competent. Obviously the real world is filled with bugs of this nature, but helpful to at least recognize where the problem stems from.