Output of @interact disappears when exporting Jupyter notebook to HTML

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This is not really a sage question anymore, since the interact feature is now realized by jupyter widgets, which exists outside of sage as well. That's good, because there is a larger community that may be able to help you.

If you go to "widgets -> embed widgets" you are offered copy-pastable html code that produces exactly the image that you want. So you get html that does what you want.

I haven't been able to find a tool that automatically does the substitutions for you. In fact, this thread (or rather the lack of a satisfying response) suggests this might still be missing functionality.

In general, the ipywidgets documentation is probably the way to go. As mentioned there jupyter-sphinx or nbsphinx may offer the functionality you are looking for.

EDIT: (elevating one of the comments below for better visibility)

The route "File -> Download As -> HTML", after one saves the widgets state, still fails. However, as suggested in the Plotly thread linked above, it can be successfully done through the command line:

$jupyter nbconvert --to html test.ipynb