All the things that you mentioned are already possible and have been since a decade or before.

• It can tell you it's done, so you notice the notification after a couple hours instead of finding musky clothes a day or two later

There's an estimate and a timer and the machine let's you know when its done. This is a 2014 model to say the age.

• It can tell you it's time to do the self-cleaning cycle

This is also something that is already automated and the machine does itself and can do manually if you ask.

• If you're doing something unusual, like washing reusable diapers which need extra rinse cycles, you can upload a custom program for that to the washer

You can add extra Rinse cycles for specific clothes and have custom programs too.

• Even more unusual, someone had stacked units and was too short to reach the buttons on the washer, so they liked to start it via the app instead of getting a stool

The controls being an inconvenience I can give you that. But, barring that its not stone age you would think it to be.

Fully automatic machines without WiFi work fine and don't need any software updates to begin with.

…which is why we shouldn’t be using WiFi for IoT devices when we have Zigbee and Z-Wave. I’ll never know why they’ll insist on using the more expensive and always changing WiFi standard.