A typical use-case for the AI Chat sidebar, at least for me, is to open it, chat, close it, browse a bit more, go back to the sidebar, chat some more.
However, as soon the sidebar is minimized the web page is closed and your chat history is lost.
On a small (13") screen, the side panel takes up a considerable amount of horizontal space so it is not convenient to keep it open.
But more importantly, the current behaviour is not compatible with how all these AI Chat providers work now - you keep longer running sessions to research something or to work on something. The current implementation works for "one shot and forget" questions but not for AI work that could potentially take lots of back and forth and could span hours.
The AI Chat sidebar should be more like a tab where it keeps state and stays "alive". Right now it is easier to interact with Claude, ChatGPT, etc. simply by opening those up in a tab and pinning that tab.