Is there any way to get from Teaneck directly to The Bronx (ie over the GW) by bus or other public transit?

Question

...it seems ridiculous to have to go through Manhattan!

Answer 1

No, there isn't.

Most public transit in the area is based on Manhattan because that is where the major trasnportation facilities were built (Penn Station, Grand Central, PA Bus Terminal - 41st St, PA Bus terminal -175th St/GWB).

You'll just have to come into Manhattan and then take the subway to the Bronx.