Unfortunately this feature was not offered to those using Chromecast to watch Netflix, at least not until now where an update to the app will change that. In the latest update to the app, users who stream Netflix via Chromecast now have the option of skipping introductions via a button that will be a the bottom of the app. It is unclear if this also applies to recaps, but hopefully it does.
This isn’t exactly the most groundbreaking update but it is a convenience and also one of those quality of life updates that will make using the app and also using Chromecast devices more viable. Users will need to be on the latest version of the Netflix app if they want to take advantage of it, so update to version v6.3.0 if you haven’t done so already via the Google Play Store.