This is just another fluff piece without too many rational arguments, based on a completely irrational fear of HFT. Arguably, HFT and hedge fund have made the markets much more micro-efficient in the past decades (as wittnessed by the difficulty of finding new arbitrage opportunities); we have to thank the central banks and big institutional investors for macro-inefficiency.
> one that applies a tiny tax rate to an array of transactions and is split between buyers and sellers
I think the current situation, where exchanges basically "tax" those who take liquidity and reward those who provide it (i.e. the fee of a market order is split between the exchange and the person who entered the limit order) is much better at providing stability to the market.
> which generate windfall profits on small and fleeting differences in prices at the expense of ordinary investors and market stability
Well, the idea behind is that prices (e.g. same stock on different markets) should converge. If you don't want that, by all means, tax the market.
> The burden of this tax would be concentrated at the top, because that’s where the ownership of financial assets is concentrated
Nope, it doesn't matter where ownership is concentrated. What matters is where trading is concentrated. E.g. HFT or every time you get paid and invest a portion in your income in a stock index-backed pension scheme.