There were plenty of three-button mouses already in the early 80's.
Of course the third button wasn't the more or less standard wheel as it is now, but simply a third button.

Anyway, what people seem to forget is that in DOS you needed to load all device drivers, if you wanted use those devices. Simply calling those as an OS function, like in Windows 95 and later versions, wasn't the case.

So if you wanted the play with a mouse, you created a bootdisk that loaded the mouse driver.
If you wanted to play with a three-button mouse, you needed a driver for that three-button mouse, and obviously the game needed to support input from that, which none of the games I know did.

So playing without a mouse was simply more convenient.

Duke3D as well. It introduced strafing as a common pattern, and really added the 3rd dimension (Doom was "barely vertical"), but it was still designed for 100% keyboard. Mouse was an option, but rarely used.

Also, that "Dwars" video has at least one pretty huge logical problem. He grossly misinterprets "designed to play without a mouse" somehow.

I've always played Doom with a mouse + keyboard, as I have found that more convenient. Keys on their own I shudder at.

(Right Hand)
Right mouse button - Move forward.
Left mouse button - Fire
Turn mouse left or right to look.

Keys just cannot compete with how easy and convenient that is.

But each to their own I guess.

I do use the keyboard for the following, etc. (Left Hand)

< - Slide Left
> - Slide Right
? - Go Backwards. Still not working for me, though instead of a 'no result' in my library, search now just hangs for both 'Doom' and 'doom'.

The result for 'Wolf' is pretty much instant.

I have something like 1668 game entries in my library now, not that that impacts with 'Wolf'.

'Doom' used to work for me, as I well recall. But not now for some weird reason. I do have other non Doom related games with 'doom' in the name. So no doubt some kind of bug is involved.

EDIT
Just discovered a leading asterisk (wild card) solves the issue. I'd tried a space either side with no result, but * does the trick.

Somebody farted!