But luckily most of the contamination happens through cough, so that these people that don't show symptoms are also less contagious.

The Mybusters really do a great job of explaining contamination visually in this segment. https://youtu.be/k1j8bh8_O_Q

You don't want to be in the same building as a sick person, let alone anywhere near them in a smaller space like an office or restroom.

That video doesn't cover aerial contamination. The kind it shows is just hand-to-hand, then hand-to-face. Don't touch your face, and the germs stay on the outside of your hands, far from your mucus membranes. I've had very good results from this.

And some case studies of disease clusters have strongly suggested asymptomatic transmission. Asymptomatic individuals who spread it to a bunch of family members despite having no symptoms, then tested positive for the disease themselves once identified.

There's no containing it at this point, the best you could do is slow it down until there's a vaccine.