You are correct. there seems to be serious limitations in the options available for the Agent icon. connecting to a web portal is not an answer and other products like Labtech and N-Able provide very easy and customizable menu options. including providing the end user the name of the computer IP address Domain and links to things like chat etc. Kaseya need to do this as well.
Andrew, please shoot me an e-mail at email@example.com. I'd like to dive into the extent of what customizations to the agent menu is needed from your perspective. Cheers.