A good question!
I don't think we need the Google Plus or other social sharing buttons on the Google home page, when the Doodle is live. But I do think adding them to the Google Logos gallery makes sense.
Maybe something like this:
While they are at it, they should document how many clicks each Google Doodle received. Would be awesome to see which are the most "popular" Google Doodles.
Forum discussion at Google Web Search Help.