In my device manager, I have
Other Devices - Tinyscreen+ with an exclamation mark
and
Ports - Teeny USB Serial with an exclamation mark

In Arduino IDE I updated as per instructions but Tinyscreen+ isn't on the list of boards.

What's going wrong?

What OS version are you running?  If on 7/8 you will need to install the drivers: https://tinycircuits.com/blogs/learn/171313479-tinyscreen-windows-7-8-driver-installation

Did the TinyScreen+ addin get loaded successfully?  (Sorry, I know you said you installed as per instructions, but it's two steps to install the addin which almost caught me out)