On the controller screen inside the Netflix app, there's now a button to skip the introduction at the bottom of the device screen. It's unclear as of right now whether the button will disappear after some time or remain present on the controller, but seeing as how the Roku and Android TV buttons don't persist, it's likely that this one will go away as well.
Both 9to5Google and Android Police reported the button's existence on Android. We tested the Chromecast controller on iOS with an up-to-date app; however the button did not appear. It's unclear if it's still rolling out, or this feature is restricted to the Android OS only. Still, it's a nice and useful feature for Netflix TV show marathons.