You *SHOULD* see all the messages from the kernel that end up in the log files you can view with "dmesg", and you should even get a login prompt at the serial port.
The "simple debricking" sticky has a link to the K3 debricking thread, which contains this video that shows what you should see during a restart (including output from the kernel):
Also, on some kindles, it helps to keep the USB cable connected to the same host PC that the serial adapter connects to, to supply a better ground connection. This seems to be more important on the K4 and K5 though...