I don't know about that particular example, but refrigerators with ice makers are an obvious case of people buying appliances way more likely to break down.
In my case it wasn't the ice maker itself, but the plumbing leading up to it within the refrigerator. I finally ended up turning off the water supply after my DIY repair broke a second time.
You can save a lot of money on almost anything if you're willing to get your hands dirty and do the work yourself. I don't think it invalidates the argument.
You're totally right. I was trying to weigh the convenience and time savings of automatic ice for 12 years vs the low cost in time and money to keep it working, but I only cited the latter in my response.