The hypothesis being discussed is "serif fonts are more readable". This article is saying there is no evidence supporting that hypothesis.

Which doesn't prove neither disprove it, so we're in exactly the same place.

Well, no. Now we know what we don't know, rather than assuming we do know something. And when we specify one font over another we can say honestly "Because I like it" rather than dishonestly "Because science tells us it's best".

I think the myth is that it's been proved, when it hasn't.

The bit you elided said this:

"The embarrassing truth is, there's no solid research to back up that claim."

Huh? Prove that a myth isn't true?

Shouldn't the burden of proof be the other way around? Otherwise it follows that goblins and fairies aren't necessarily false beliefs.

You could imagine anything and claim it is not necessarily a false belief until someone proves that it is. Worse, you could imagine something with properties such that it is impossible to prove that it doesn't exist. At this point we'd all have to go around accepting your imaginary world of "not necessarily false" beliefs because we can't prove them to be false. What a terribly unstable situation that would be. Imagine, millions of people believing in one not necessarily false set of beliefs, and millions of other people believing in a different set of not necessarily false beliefs, and both groups claiming that the other must be wrong, each without method of proving the other to be false. What a dangerous way to reason about things.

No, what he means is that the author should have said: "It is a myth without proof" but not "a myth which is not true", which is what the last sentence says.

Myths can be true, like, you know, the existence of Nineveh (yes, it was a "Myth" and then it was proved true).

I just flipped a coin at my desk, which I'm covering with my hand. There's no evidence to support that the coin came up heads, so saying that it's heads is a false belief. There's no evidence to say that it came up tails, so saying that it came up tails is a false belief. Since the beliefs that the coin came up either heads or tails are both false, we can determine, by process of elimination, the the coin must have landed on its side.

The myth is a false belief because there's no evidence to support it. However, being a false belief doesn't prove the statement false, simply undetermined.