For a customer that just doesn't get it, I don't see why the dishwasher/vacuum cleaner analogy is bad as long as it's delivered nicely.
Analogies don't convince and are often seen as condescending or evasive: "I asked for help, and I get word games"
I'm not sure of the best way to convince a customer (who is wrong) that is questioning the rep's expertise. Provide more data and explain differently, I guess.
It's also bad because the solution was actually pretty straight forward send them the support docs (or the support contact) for their website. At least then you're pointing them in the right direction.