Appliances have potentially serious failure modes that don't involve battery fires. (We had one here a couple of weeks ago, which would have flooded out our basement if I hadn't been able to cut power to the pump involved.) Being able to cut the power completely and instantly is not negotiable for a lot of appliances. I wasn't even taking battery fires into consideration when I wrote about failure modes—I was talking about things that already happen to plug-in appliances right now.

Yes, the added weight and complexity are likely not all that significant here, but they're sufficient that, even without the power-cutting issues, they outweigh any benefit of attaching a battery to the appliance directly. It's just not a particularly useful idea when you get pretty much the same benefits with none of the downsides by incorporating the batteries into the building's power system separately.