I have done as you suggested. My kindle says I have bought a book and it has been downloaded, but it does not appear in my list of downloads.
Make sure your Kindle is connected wirelessly. If you are not sure, go to Check the Status of Your Wireless Connection.
Note: Select Pending Deliveries to view titles that have yet to be sent to your device or reading app.
Your Home screen may not be displaying all of the titles downloaded to your device.