If anyone is interested I have pinpointed the problem by flashing an older version of Android. These tests were done on my 16GB Nexus 9.
With a freshly flashed version of Android v7.1.1, with all user data wiped, the Kindle app hangs when you click the Start Reading button.
With a freshly flashed version of Android v7.0, with all user data wiped, the Kindle app runs normally. Clicking the Start Reading button allows me to enter my Amazon account details and start reading.
If I let my tablet download the updates to v7.1.1 and install them, without wiping the user data, then the Kindle app still runs normally. So the app does run on v7.1.1.
But ... if I now delete the Kindle cache and data from Settings/Apps then the Kindle app once again hangs when you click the Start Reading button.
So the problem is specific to the v7.1.1 version of Android, but the problem is limited to whatever bit of the Kindle app configures your Amazon account.
In theory there is an update to v7.1.1 on its way to my OnePlus 3 phone, and when that arrives I will try the same experiment to see whether it is a device specific problem.