Flyout: use the new integration point

[humitos]

Use the documented integration point for our theme.
Read the official documentation at https://docs.readthedocs.io/en/stable/flyout-menu.html

I'm thinking about simplifying how all these integrations work and use only the ones that are documented. The idea is to inject the documented HTML tag that will be replaced by our integration at serve time.

Besides, I'm populating the template context with all the READTHEDOCS environment variables, so we can stop passing them from the Read the Docs application at build time by injecting them into the conf.py file.

Related: readthedocs/readthedocs.org#10127

[humitos]

Besides, I'm populating the template context with all the READTHEDOCS environment variables, so we can stop passing them from the Read the Docs application at build time by injecting them into the conf.py file.

Most of these "context" variables should be converted into environment variables that all the doctool/theme authors can benefit from to integrate properly with their doctool/theme. Note this is something risen at readthedocs/readthedocs.org#9423 as well.

[humitos]

I'm closing this PR because this integrates "the old flyout" and we are moving forward with the "new flyout out" now, see #1523