One often hears Jewish people remark, “We Jews believe that we can come directly to God.  We have no need for a mediator.”  Is this an accurate representation of historic Judaism?

Not exactly.  The individual Jew does not approach God directly.  This was recognized in the early 1950s by Will Herberg, one of the most prominent Jewish thinkers in America at that time:

In both Judaism and Christianity . . . there is no such thing as a direct and unmediated relation to God; this relation must in some way be mediated through one’s covenant status.  In Judaism, however, it is by virtue of his being a member of the People Israel that the believer approaches God and has standing before him; in Christianity, it is by virtue of his being a member of Christ . . . To be a Jew means to meet God and receive his grace in and through Israel; to be a Christian means to meet God and receive his grace in and through Christ . . . Authentic Judaism is therefore Israel-centered . . . while authentic Christianity is Christ-centered.  In neither need this centrality lead to a diversion from God, because in both it is through mediation that God is approached.