Actually the exact opposite will happen.
Only the HFTs will want to move away, the rest will want to move to such a market. For the rest of them, operating in a market that allows HFT is pretty much the same thing as operating in a market for lemons: https://en.wikipedia.org/wiki/The_Market_for_Lemons
Asymmetric information in markets means 'the bad drives out the good'.
Just as running an honest used car market will attract buyers and sellers, running an honest stock exchange will attract traders.
This is what I can't understand about this stories, and makes me doubt the claims against the HFTs.
You should consider asking yourself why most such trading networks actually go to great effort to encourage HFTs to trade on them. (Hint: no liquidity no customers.)
And it wasn't just Tradebot doing it. There were many firms already active with similar styles in the early 2000s like GETCO, ATD, Madison Tyler (now Virtu) and a bunch of smaller guys. HFT does not depend on Reg NMS, fragmentation, or rebates. HFT firms are super successful in futures markets with a consolidated limit order book and fees for both sides of the transaction.
I'm not sure what you mean by "one flat rate." Exchanges already charge a flat rate for connectivity (the rate is per month, usually a couple thousand bucks or so).