Why do we not use both E and B fields for a receiving antenna?
The premise of your question is incorrect. Receiving antennas "use" both the E field and the B field to develop the voltage at their terminals. Antennas constructed from straight conductors are most sensitive to signals whose E field is parallel to the conductor (inducing a voltage) and B field is perpendicular to the conductor ("cutting across", which also induces voltage).

Loop antennas are a special case. They depend almost entirely on the B field alone, and are not very sensitive to the E field at all.