Thanks for your insight. I see, that does make Stanford seem like a better option then.
Although the reason I valued writing a formal paper is for the computer science phd application setting. It seems that applicants have numerous first author papers written in top conferences. Surely these are not all from formal opportunities/thesis, but I worry that if I were to attend Stanford and research under an advisor but not publish/finish a formal paper then that would make me less competitive - even though I would have a recommendation letter from them. Perhaps this is unproductive overthinking however ...
Would appreciate any insight !