Sheeet, I don't even own a mouse. Haven't for at least 10 years now. Keyboard-only is my only real option, so that's how I play. And it is harder. And I also choose not to use saves, but that's for other reasons entirely. And of course no mouse means no freelook, thus another reason for me to stick with pure vanilla maps.
Funny thing about DOOM is it came out at a time when you couldn't guarantee the user would own a mouse. A lot of people were still running straight DOS, even if they did have Windows 3.1 or whatever (RAM wasn't cheap, had to run bare-bones to play most games). But mice weren't that expenive either, and some games did use them, or at least had an option for them. So really, there was no "default" control for DOOM, you could go keyboard-only, keyboard+mouse, or keyboard+joystick.
But never FPS (including Quake) became almost impossible to play without a mouse. The vertical autoaim still works to some degree (turn off mlook, and adjust sv_aim as needed), but it still makes the game difficult (instead of impossible) and is almost completely useless for DM.
The closest you'll get to the original is with one of those three input modes, 320x200 software rendering, and either play it in vanilla or Chocolate Doom. I also run it with Freedoom, but then I run everything with Freedoom. Theoretically this doesn't affect gameplay, but some of the sounds and sprites are either missing or underdetailed, so it makes the game a bit harder. For example, some of the revenant's sounds are empty, and his sprites are almost entirely black, so he's quite the ninja bastard at times (fades in really nicely against ashwall, in darker corners, etc.)