I fear calling the web a public square, thinking that is is truly open, is a myth. All of the infrastructure upon which the web runs is owned by corporations. The technology upon which the web is built is intended to be open, but they are under control of corporations.
Corporations pushed the web towards HTML5 in support of apps that they utlimately control. Corporations own the browsers that we use and with coding changes can force all of use towards a direction they want. The network infrastructure upon which the packets ride are owned by corporations who can, and have, allowed governments to capture and review those packets.
Sure, you can host content on a web server running in your house. What happens when Comcast or AT&T decides to not sell Internet access to you? What happens when the coporations that control DNS remove your domain from their servers, or direct requests somewhere else?
None of this is truly open. All of it is at risk if it does not serve coporate interests. Right now the best we have are degrees of openness.