I agree that finding the online doc is slight pain.  It is not listed in the Modules index because idlelib, not IDLE, is the stdlib module, and does not have a doc page.  I may add one or, if possible, make the page serve as one, as part of this issue.  'IDLE' *is* in the Index, and the first link is the correct one.

I don't control and would not unilaterally change the main page.  I am hesitant because I expect that there are people who would think it unfair or deceptive to especially promote IDLE this way versus other stdlib packages and non-stdlib IDEs and editors.  I might subscribe to and post on python-ideas to see what other users think.

Help => IDLE Help displays an offline copy of idle.html as of the release date in a tktinker text window.  Have you noticed that?  Would "IDLE doc" be clearer?

I could also add label with the online url and date above the text.
[https://docs.python.org/3.x/library/idle.html as of <release date>]