That, or the "home server" component will be a separate mode / augmentation for what are still primarily cloud-based services.
I'd much rather locate my personal server (or VM) in a real datacenter and behind a better SLA than my flaky Comcast.
I think you're still right, though: the larger picture these things will slot into is as subsidiaries for what IBM has termed "cloudlets": automatic device-launched containers, operating on ISP commodity-IaaS clouds (i.e. at the "edge" in the same city the user is in, where all the CDNs and caches and mirrors they access already are.)
Cloud services would probably be written with cloudlet service-worker containers in mind, and then the home-server hardware would just be a latency-optimization, "stealing" workloads from your cloudlet to your LAN when all the traffic is just going to be over your LAN (e.g. for a home streaming service like Plex, or a game server at a LAN party) and then moving them back if you try to speak to the cloudlet from outside your LAN.
Cloudlets are the more important part of that vision, but I could see the home-server part of the design coming first, just because so many companies seem interested in putting devices into home networks that could run them (e.g. the Apple TV and its class of products; the Amazon Echo, Google Home, and other "voice boxes"; most "smart home" device hubs.)
What does the ToS say, though?
This is the result of checking my connection for IPV6 compatibility:
The thing I expect these to lead to is for everyone's router to become an implicit home server, and for that to lead to changes in the way we write "cloud" services.
Right now, a lot of background "checking up on things" for services like group chat apps, happens directly on people's phones, because they're (almost) always on and (almost) always connected to the Internet.
Now imagine that it's guaranteed that everyone in the world has an implicit personal (or family) server their phone can communicate with, and can automatically reach over and launch processes on. All the background logic would move to the server. Phone apps (and a lot of computer apps) would become simple thin-clients that talk to your home server.
It'd be like IRC bouncers, but for everything, and for everyone.
Start a multiplayer game? The lobby server launches automatically on your router.
Install a "free VoIP service" app? You've actually just installed an Asterisk server on your router; the app just configures it and acts as a softphone for it.
Want a blog? The button on WordPress.com doesn't start up a LAMP container in their cloud; it starts a LAMP container on your router.
Etc.