tldr; How you use the computer / manage application windows is more important than multiple screens for productivity. If the way you interact with windows and applications isn't cerebral then you're doing it wrong.

First of all, I'd say his argument is perhaps less applicable if you're a programmer. If all you're using a word processor and a browser, then maybe you can get by. If you're doing programming and designing, and need a couple of applications as well as a few terminal windows — then maybe it's not so applicable.

However, I do find this subject matter really interesting. I've been thinking and observing about this a lot just this last year, and I think it's not so much about screens as it is how poorly most people use computers.

I think the main productivity killer is the cursor and how people do window management: focusing an app, resizing, moving, shifting screens and so on.

As an example, say you as a professional need to use three applications, and most of the time you use these applications in a maximized state — taking up most of the screen space.

Now, if you want to shift attention between these applications, there are a couple of alternatives. You could alt-tab, which is imprecise and slow, since you have to mentally go through the alt-tab stack of applications and find the one you're looking for. Of course, you can use alt-tab to toggle between two specific windows, but if you slip up and press it twice then you bring focus to some other application.
It's just not really a good way of switching windows.

Another way is of course to use some graphical element targeting a specific application. Maybe the bottom menu (the dock on OSX). This is also imprecise and tedious. Windows tries to fix this with having the order of the applications correlate to shortcuts. So the first one can be activated using Windows Key + 1, the second one with Windows Key + 2 etc.
This still leaves you with the problem however that you lose state between toggling applications since your cursor isn't where you left it for that particular application. There's also not a good way of toggling between windows for an application: say you have two chrome windows, and you want to switch between them.

Or maybe you have multiple screens, and you now have to drag the cursor between the windows. Super slow and imprecise, and maybe the cursor even gets caught between the screens since one of them is in portrait mode and the other is in landscape — now you have to spend even more mental effort just to get it to where you want to go.

The way I've solved these issues — which has given me a huge productivity boost with a laptop in particular but it's also very tangible on desktop computers with multiple screens as well — is to create a really smart system to jump between applications where it remembers where the cursor was for that particular window, and if the cursor is in some nonsensical place when you jump to an application window then it's instead reset by centering on that particular window. It also uses a graphical element surrounding the cursor which shows for about half a second, so I don't have to spend any mental effort to find it.
I also have a very nice system to manipulate windows which sort of works like Divvy where using the keyboard I can on the fly define a grid on the screen, and move or resize the focused window inside of that grid.

Using things like this makes it cerebral. I don't have to think, or spend mental effort targeting things. It's all super precise and I don't even think about windows and screens most of the time, it's all mechanics that have become invisible to me when I work.

I'd bet with this setup I'd be a lot more productive with one screen than someone with two screens without a similar setup.