I played OG Doom for years with keyboard only. Up/Down arrow for forward/backwards, Left/Right to turn, and then holding Alt to strafe. RIght hand on the arrow keys and left hand controlling strafe/run/weapon selection.
I don't honestly recall if mouselook was even an option back then, but I tried playing with those keyboard only controls recently and literally could not even. It's kinda wild how hard it is to imagine anything other than WASD-mouse for a FPS these days.