Right, but imagine something like a Raspberry Pi with some cheap storage attached. It could collect data from your house (temps, duty cycle of furnace/AC, solar panel power, garage door state, collect your photos/docs, receive email/IM asynchronously, etc). Sure you'd want offsite backups, maybe at a friends house... if his server had a public IPv6 address. With IPv6 it would be much easier to remotely use all those resources/files.
You could give friends/family access to your photos, you could receive/send messages even if your phone/laptop was down, off, stolen, or gasp without a network connection. You could share files, and if someone wanted to confiscate it they would have to come knocking.
IRC never did async well (you have to be online at the same time) and never became popular with the general public. Skype did p2p well, back when most had an IP address, a physical network cable, and left their machine on (unlimited power). As NAT and mobile devices (that couldn't constantly be available for bouncing around a NAT and participating in a DHT). As a result Microsoft centralized it, so low power clients could connect as needed.
IPv4 basically turns a home computer into a consumer of internet services, even for silly things like using your cell phone to open your garage door requires a cloud service to be involved.
My main point is many traditional web services could be provided by the individual. You can share docs, communicate, and manage your increasingly complicated home environment easily with IPv6. Sure some would still use centralized services, but many more would do it yourself.