There is no such thing as public property, it is a fiction created by government. Go try to plant a garden on "public" property or use a government car without permission-after all its "public" property right so you have a right to use it don't you? There is no such thing as public property, the government is a private criminal gang that extracts funds at gunpoint from it's victims and monopolizes certain essential services to give the impression that is is essential for the continuance of society when in fact everything it does could be provided better and voluntarily by the market.
The public would not own a "public" exchange, the state and members of it's criminal gang would. The exchange would never be run successfully since it is not responsive to market forces, only in a market where a business lives or dies by getting money voluntarily from its customers can a business know whether what it is doing is what customers want and is providing value. In a state system where the money for the service is stolen at gunpoint, there is no way to know whether the service provided is working or is something people want since people cannot withdrawal their funding support.
Democracy is a tyranny, it is 51% violently forcing their will on 49%. That is not persuasion, that is not negotiation so that both parties feel served, that is violence. Democracy is evil.
The nature of bitcoin is to be decentralized. What you should advocate for is a decentralized exchange that avoids the problems we saw in gox because its decentralized. The challenge is to determine how to get fiat into the exchange quickly and digitally without using the traditional system.