Sumner showed the EMH predicts exactly this, for a certain distribution of information and returns.
Assuming you are only talking about the blog post you linked, I can't see any sort of proof there, there are some hypothetical numbers about Bitcoin plucked out of the air, and he then goes on to assert that EMH is true whatever happens. If a theory is true whatever happens, it is not a useful model for thinking about the world.
If you take the EMH to mean that prices perfectly (or near perfectly) reflect information (public and private) about an asset, I'd say that mortgage CDOs (to pick a historical example) provide a perfect example of prices NOT reflecting all information - Goldman Sachs insured them with AIG knowing that the prices were incorrect days before the bust, but the price didn't reflect this. Given the incredibly broad claims made by EMH (market prices efficiently reflect all information), one counter-example of corruption, euphoria etc etc driving prices rather than underlying value is enough to disprove it, if it is a meaningful model of markets which can make useful predictions. That is why people cite booms and busts as evidence against EMH, because sudden changes in value are often completely unrelated to the consensus price as measured several days later or before, they are caused by panic, irrational behaviour (in aggregate, not just individually), or insider knowledge; information is so unevenly spread in markets, and manipulation pays so well (e.g. LIBOR), that they are nowhere near efficient.
I'm confused. Sumner showed the EMH predicts exactly this, for a certain distribution of information and returns. How is the real world agreeing with Sumner's model evidence against Sumner's model being correct?
What is dangerous about the EMH to me is that it encourages a sort of worship of markets
The EMH is simply a rule which allows you to turn a statistical distribution of new information into a statistical distribution of price movements. This is what Fama did to get the Nobel prize. A corollary is a "speculators can't make money except by chance" theorem, or in it's weak form, "technical traders can't make money except by chance".
It's not a "markets are benevolent wizards that solve all the problems" theorem. It's not a "no recessions ever occur" theorem. It's not a "prices will never change suddenly" theorem - in fact, it proves the opposite of the latter for certain information distributions. If you argue against these latter claims, you are not arguing against the EMH.
it needs to be proven in the first place
That's not how science works. You prove theorems, the best you can do to theories is fail to disprove them.