If you try to design a system of publicly traded companies from first principles, then the purpose of exchanges should be to provide a service to publicly traded companies. That service would be maintaining the authoritative record of who owns how many shares of the company.
I don't think there's any reason to think that exchanges' main purpose for existence is to provide service to publicly traded companies. Exchanges are just a store, and their purpose is to buy/sell stuff and make a profit. So, in that sense, having multiple exchanges selling the same stock is the same as having multiple bike stores selling the same bike...it's competition and there aren't many downsides to it.
As for the social cost, you do realize that HFT does not come for free to society. At a minimum, society must somehow pay for those well-paying jobs to developers. And what do those developers give back to society? Before you answer, please consider whether you should distinguish between algorithmic trading and HFT. I do believe that algorithmic trading is useful to society (because it can do the job that screaming traders used to do better and cheaper), but the subset of HFT is not useful.
Nothing comes for free to society. Funding a never-ending number of failing start-ups doesn't come free to society either. What is that giving back to society? Your implication is that HFT trading is somehow taking something from society that it doesn't deserve. What is it taking? Also, in terms of making distinctions: HFT is the superset and AT is the subset, not the other way around. All AT shops are HFT shops, but there are actually quite a few HFT shops that don't run algorithms and instead run based on inputs controlled by traders.
The only things anyone ever seems to be able to answer for HFT two-fold: one, that stock prices change faster, and two, that the spread is smaller.
This is the crux of the whole thing. Why does anyone have to answer for HFT? This is implying that HFT is hurting people however you haven't not given any proof of this (besides that it doesn't add value). If anything, I think HFT's biggest crime (as someone else has mentioned) is that sucks some really smart people into working on problems of a rather limited scope.
The reason there are multiple exchanges is the same reason you can buy the same pair of shoes at 10 different places: it is competition.
Also, what is the social cost you are referring to? There are a lot of people here saying that HFT doesn't provide "social value". Does it somehow provide less social value than traders screaming at each other in the pits? Similar types of trading always existed, except now it's done by computers. For what it's worth, HFT provides well-paying jobs for a lot developers and ops people and often is very supportive of the engineering community.