Firstly, you have to understand that Amazon is actively chasing out the last opportunities people have to download a copy of their recently purchased books.
The removal of the "Download and Transfer" button on newer Kindle models and the increasingly difficult process of downloading content through an old Kindle app, are just marks on the wall telling us that Amazon truly doesn't want us to download the books we paid for, and that they will do anything to shut down any opportunity of ownership.
If you understand this, and you still want to buy books on Amazon and never backing them up, then don't act surprised when your account gets shut down for "suspicious activity" and realize all the books you paid for are lost forever.
Amazon does not want to see you owning a book.
For this tutorial, you will need:
Go to Calibre, download and install the "KFX Input" plugin from the plugins menu.
Unzip the deDRM file. Go back to Calibre > Preferences > Plugins > Load plugin from File and then pick deDRM_plugin.zip, after that, restart Calibre.
Open up Preferences again > Plugins > search for deDRM > Customize plugin > "Kindle eInk ebooks".
In this window, you will need to type out your Kindle serial number which you can find on either the Amazon website (Manage Content > Devices > Your Kindle name) or within the Kindle menu settings (Settings > Device options > Device Info).
Now you will have to download your purchased books on your computer. Go to Amazon's website > Manage Content > Books.
Unfortunately for us, there are no ways to download our books in mass so you will have to download each one of the books. (Note that books you got for free don't have DRM (azw3 books)).
Once downloaded, you will drag the book to Calibre. Do not use the "Add books" option.
The de-DRM process will run automatically and once it finishes you will be able to read and edit your book. In case your book can be read but remained as a KFX file, you will want to convert it to EPUB, for this you will also need to install the KFX Output plugin.
There is a chance this won't work for the first time, in case this happens make sure the Kindle is correctly registered, you typed the full number serial without extra spaces and the deDRM plugin is set correctly.
You also can investigate if the deDRM plugin is still working on with newer books with a quick search check on Reddit or Mobileread, I know you can research things.