The Doom demo is essentially the PC version of the game set into arcade mode, with options of a Razer Keyboard and Mouse or a Razer controller. Upon researching, I could not find any real evidence or reports that these items are particularly infamous for built-in input lag and all three input devices were seemingly wired. It's not an exact measurement, but swinging your aiming reticle around the screen is not instantaneous, and anyone that has played Doom before can instantly feel the difference.
More to the point, I was missing shots, and it was initially difficult to time melee hits against enemies. It's not that the input lag makes Doom unplayable, but it makes it harder, and it makes you worse at the game. It's the kind of thing that would make you reboot your console and check your TV settings.
The thing is, after a few minutes of playing, I was still conscious of the difference, but it felt like it mattered less. It was like controlling that big gun in some shooters with the swimmy reticle that dragged behind the input. I knew what was happening and eventually my brain and my hands compensated for that difference. Was it ideal? Definitely not. Was it a way to play Doom in a stream without a console or expensive GPU? Approximately. And that will probably be enough for a lot of people.
In the battle between the perfect vs. the good, Stadia's input lag is definitely not perfect, but it seems like it can be good.