It's not, but most people (or at least, most people who grew up with computers) can type faster than they can write, so the operative comparison is between frantically writing and typing at a comfortable pace.

I've never taken notes on a laptop. For those who do, are you typing every single word that the speaker says?

I've taken notes on paper during many, many talks. I don't write down every word that's spoken.

Isn't the cognitive load likely due to to deciding what content to record, not due to controlling the brain-hand-pencil-paper interface / brain-hands-keyboard interface (as it were)?

The article is saying that typing the speaker's words verbatim is exactly what most people do when taking notes on a laptop, even when they are explicitly warned that it's a bad idea, whereas it's generally not possible to do so when writing notes, without some kind of shorthand.

In any case, that's not the point: regardless of what text you decide to record in your notes, it's faster to do so by typing it than writing it for most people. That means that if you switch from writing to typing your notes but continue taking notes in the same style, you should in theory have more time to figure out what to write without falling behind the speaker or needing to rush your typing.

I take notes on a laptop for school and what I have learned is it is very valuable to reorganize notes after readings or lectures. My goal is always to make a document that I can review later that is structured. Restructuring online without rewriting entire pages is pretty efficient. Also, with some discipline you can tag sections and grep them later.

A disadvantage over writing is not having multiple columns if you are using plain text. Tools like onenote solve this, but I find it too distracting to use in general.