IMHO no. Every chatbot has so much wasted space, it really doesn't need to be full-width. Also, what's easier?

Option 1: Being on a tab, copying the URL of the tab, switching to the chatbot tab, pasting the URL and writing some instructions about what to do with that tab.

Option 2: Clicking on the "summarise page" button (whether from the sidebar or from right-click context menu), and having the browser pre-fill the prompt with the URL + the reader view version of the content on that page.