Thank you Blossom, you're so right!
I noticed that my old Kindle had lots of problems turning on and off, but I thought I didn't hold the button long enough (basically, that it was my fault).
Turns out that my new Kindle turns on and off much, much better (about 10 seconds faster!) doesn't have the flashing issues, and doesn't hang.
The old one is in its case, ready to be shipped to a competent doctor.