I've been thinking about this a lot lately, lumped under my own umbrella term of "communal computing". Instead of one large instance, many smaller instances hosted by independent administrators and often targeted toward particular communities. Avoiding the scaling issues of the former type of service by shrinking things down to where individual operators can run them. Examples include Mastodon, shell services like tilde.town or sdf.org, old-school MUDs, chat servers.
As an aside, this will happen on all businesses too, not just web sites. As software, services, and robotics get better we will start to see solo operated versions of every business.
Once we start dropping the average corporation size below 1 (each person runs multiple businesses) most of capitalism’s ill effects will be blunted.
Mastodon is leveraging forces from many different directions: first by building towards the existing GNUSocial protocol and its userbase, and then as different communities come in, getting donations of their time and skills to operate instances tailored to their needs. Businesses have angles on this too: for example Japanese art site pixiv made their own instance and client app. The federated model allows some divergence but also the "loose ties" of a common protocol. There is the prospect of embrace-extend-extinguish occurring, but this historically only happens some of the time, and it requires the full pressure of a giant tech firm and all their marketing strength to make it work.
I launched my private instance and started from there. I can say I'm very surprised, the system works very well, never had major problems and the communication with people with accounts on other instances (since I'm the only one in mine) works flawlessly.
This is how the Internet was supposed to work (not full of walled gardens). The good thing is that we're seeing some great decentralized applications and tools appearing recently and a renewed interest in this area, which is great.