No, that's not the same thing. The law doesn't say that you can't have any pirated content on your service. It says you can't knowingly have pirated content. Once you learn about a stolen movie on your site, you have to take it down.
When, like Dotcom, you instead actively solicit pirated material, optimize the pirated material on your site for quality, and pay users to put more pirated material on your site using a sliding scale based essentially on how pirated the material is, the DOJ reasonably concludes that you are not a public video site like Youtube, but rather a criminal conspiracy.
Dotcom appeared to have been operating under the misconception that a "notice and takedown" system insulated him from that charge. Dotcom didn't actually read the DMCA, because notice and takedown is only ~half of the responsibility of an online service. The other half, written right into the DMCA, is operating without red flag knowledge of pirated content.