My Kindle is not one of the new models, but the operating system on the Kindle is the current one. I just checked and it says it is up to date and running OS 5.18.2
I just double-checked the behavior I'm seeing. I tried it again on four different books, with the "Downloaded" filter on the Library view on my Kindle. In each case, when I bought a book on the website on my computer, the book appeared in my Library and the Kindle showed "Downloading…" under the book title for one or two seconds as it downloaded the newly purchased book.
I did a little searching, and this auto-download after purchasing behavior seems to apply to all Kindles if a) the Kindle is designated as the default Kindle device in the Amazon account (Your Account: scroll down to "Digital content and devices" then select "Devices" and click Kindle to show the Kindles registered to the Amazon account, b) Sync is turned on in the Amazon Account and on the Kindle, and c) the Kindle is connected to WiFi.
Sync should be turned on by default on the Kindle (under Settings-->Device Options) and on the Amazon Account (on the Devices screen described above, there is a small "Preferences" label near the top of the screen to switch from the Devices list to the Preferences regarding Devices.)
It seems that if all the setting match up, this should work on any Kindle with the current Kindle OS.