I solved this. The instructions are incorrect in several ways. The amber LED only blinks if there is active traffic going to the UART ( e.g. connected via TeraTerm and watching the LED as you send a command ). So when step 6 of the "Getting Started Instructions" card included with the device the instructions says you should see the light flash as an indicator as to whether or not you need to install the drivers; it is wrong because the light will not flash at that point no matter what.
The USB to UART driver installation instructions are also out of date or incorrect. I finally got drivers working by manually installing the Win 7 versions from the extracted driver install directory. Even in those, there is no "CypressUsbConsoleWindowsDriver as Figure 4/Step 9 shows, nor is there a setting for "Enable Port Persist" in the USB Serial Port driver properties ( steps 10 and 11 ). Apparently, you don't need either of these.
Lastly, and this may be my bad since it has been a long time since I have had to use a terminal emulator, but when you connect via TeraTerm, it will sit there with just a cursor block and nothing else, which gave me the impression that it was trying to connect and was hanging/failing. You have to hit return/enter and then it will respond with a prompt that says "zynq>". To be fair though, I did try connecting via Putty when I first started with the Win10 drivers and got a timeout message. Maybe my Putty telnet settings were off and the Win10 drivers are actually fine, but I don't care enough to double check since I have it working now.
Hopefully this will save other people from wasting a bunch of time like I did.
Thank you so much! I signed up here just to write this comment. I think this answer has saved a lot of people's time over the past 8 years. You saved me a lot of time. Thank you!
Thank you so much! I signed up here just to write this comment. I think this answer has saved a lot of people's time over the past 8 years. You saved me a lot of time. Thank you!