Sorry, just to understand: are you saying that it loads immediately after a refresh? Or are you saying that if you manually re-run after a refresh, it somehow happens immediately?
If it is the former, I think there may be some misunderstanding as to the way RunKit works. The code is not re-run on every refresh. The author controls the running of the code (either by hitting shift+enter, or by clicking the run button). Once the code is run, others that visit the page, or you yourself refreshing, simply shows these same results.
If it’s the latter, and creating a document without a log in and hitting shift+enter causes them to appear immediately then that would certainly be a bug. But want to just confirm which of these two it is first.