This is a good example. I think from the context you would imply that the police want people to stop crowding around, staring. In this case, you can say, “Please go about your normal business” and that means, “go back to doing what you would normally do (and stop doing this unusual thing like staring at a crime scene)”

Instead of “skill” here, I would say “the ability to.” : It’s crucial to have the ability to go about your business without being absorbed by the grim reality (of what’s going on in the news).

In this case, I think it would sound a little better to say, “…who refuse to go about their business when they see an accident. They slow down and watch [, causing delays for the rest of us].”

Great example!

Just a couple small edits, mostly just prepositions and small words: “Every morning you can see dozens of people walking hastily, going about their business in the streets of capital cities.”