For those wanting definitive confirmation, the new Amazon Fire TV 3 does allow sideloading apps like Kodi. Apart from a small issue with my Downloader app that has now been fixed, sideloading on the latest Fire TV running an all new Fire OS 6 is exactly the same as it is on other Fire TV and Fire TV Stick models.
The Fire TV 3, but more importantly Fire OS 6, has the same developer options that are necessary for sideloading apps as previous Fire TV devices. The only difference is that the Fire TV 3 does not allow ADB over USB like some other Fire TV models allow. This means that you cannot connect the Fire TV 3 to a computer via a USB cable and send ADB commands that way.
The “Apps from Unknown Sources” option allows users to easily sideload using other apps, like my Downloader app and file managers. The option is present and functions the same way as it always has. The “ADB debugging” option is also present and functional. This allows sideloading via a computer.
In addition to allowing sideloaded apps, the Fire TV 3 and Fire OS 6 display sideloaded apps alongside apps from the Amazon Appstore, just as other Fire TV devices do. If you’re having trouble seeing your sideloaded apps appear on the home screen, be sure to follow these steps.