I don't think (s)he was talking about the payment processors, but about the centralization of mining, which is a different issue.
And a payment processor in Bitcoin is closer to a company that runs cash registers than a provider like VISA. The worry is about the companies that want to control both ends, like Coinbase, not payment processors in general that simply take care of accepting BTC on behalf of the payee.
They'll up being the same in the end it's just the logical financial evolution of the scheme.
With ever increasing mining complexity coupled with ever growing demand for more and more mining since the rate of transactions in the system is directly tied to its mining rate when the block rewards will become rarer and rare till they eventually halt you'll pretty much will end up only with the "Banks" doing or controlling the mining since they'll have to ensure at least a minimal mining rate which covers their transaction volume.
the rate of transactions in the system is directly tied to its mining rate
How so? That's not my understanding at all - you could have a rate of transaction at the level of VISA with just a dozen computers or so. The mining difficulty is a consequence of having many miners (which makes the software auto-increase the difficulty of the problem), not of the increased transaction rate.
the block rewards will become rarer and rare till they eventually halt
That's why there are transaction fees, which pay the miners in lieu of the block rewards.
It's an ouroboros turning into a catch 22 you can't have an open system with high level of trust if the computation is trivial.
Ever increasing difficulty raises the bar so high that it centralizes a system due to the hardware requirements (there is no way to effectively mine BC today on standard computing hardware sorry) and their costs.
Look at the BC network now it's either huge centrally controlled mining pools or multi-million dollar mining farms which are controlled by individuals, this isn't decentralization of anything but computing assets as far the the pools are concerned.
> All of those "proof of work" type schemes eventually will dwindle down to the same centralized network they are aiming to replace.