What is being argued is that it is not out of the question to expect that such a system may or could exist. It's not unreasonable to think that it could.

Occam's Razor. Are all of our nukes controlled solely by systems that do not involve a mouse, or might there be a computer sophisticated enough somewhere in that launch process that it might make use of a mouse?

“A left mouse push fires it. Kinda crazy really. We actually asked for a great big red button, but they wouldn't give us one.” ¹

Though that's in reference to submarine-launched non-nuclear cruise missiles, not ICBMs. ICBMs might still rate a great big red button.

Not that OP's experience is any less silly.

¹ http://news.bbc.co.uk/2/hi/uk_news/magazine/3078097.stm

As I said, I agree that it is irrational to think the guy was whiteboarding an ICBM launch. (Not to mention while terrorists could conceivably get a dirty bomb, an ICBM seems well out of the question.) But there is nothing in those photos that would lead an average person to believe it was a game.