As a low-effort, low-impact change, I would add a setting with a choice between three options: 1) Always start collapsed, 2) always start expanded, and 3) remember last state.
For me personally, the last state is simply not a valid indication of how I want future searches to behave. To me, remembering this only introduces inconsistency and unpredictability. Manually re-collapsing it before closing the search results page is an extra step that could be avoided. My workflow in search results pages is usually to open links from them and move on. If I'm opening links in the same tab, I will have no opportunity to manually re-collapse the view. If I am opening links in new tabs (my default), I don't want to have to remember to go back to the results tab and collapse it when I'm done. I am not realistically going to put my actual tasks on hold to spend time micromanaging my settings with every search.
That said, I wouldn't consider a new setting a full solution, because it does not address the usability problem when it's expanded, or in the Assistant. The problem there is simply stated: rapidly moving text and links are unusable. Why should the majority of the loaded content on the page be unusable while the LLM is still working?
My previous suggestion of making the text box expand in larger chunks instead of line-by-line would help mitigate this problem (though not entirely solve it). Alternatively, perhaps there could be a middle ground between "expanded" and "collapsed", where the box is statically-sized but large enough to use, with its own scroll bar. Maybe 10 lines tall or something like that. Then you could eliminate the jumpiness altogether, and both the quick answer AND the search results would be usable at the same time immediately upon loading.
As for the assistant, auto-scrolling line by line adds a little extra friction every time I submit a query. I can scroll up and stabilize it, but I feel like I need to fight against the UI every time. Again, the fundamental problem is that jumpy text is not useful. Jumpiness should be minimized if it can't be eliminated.
Thanks for your attention. I'd be happy to discuss this further, and I hope you and your team can come up with better solutions to the fundamental problem than I can off the top of my head.