Embed Flask page in another without code duplication?
To re-use a chunk of HTML, you can use Jinja's {% include %} tag. If that's too limiting, Jinja macros are also well suited. You can define your macros in a separate file and import them with {% import "path/to/macros.html" as my_macros %}.
Flask-Assets can help with the organisation of your assets.
As for using Blueprints, yes you should use them. But they mostly apply to Python code and HTML templates are organised in a different realm, so maybe their use is unrelated here.
You can't always remove all duplication though. If your game needs to affect three distant locations of the server-generated HTML, that's bits of template code to copy in every template that includes your game.