I have no idea why the dry adiabatic lapse rate is so great in the chart. I've only ever used Skew-T log-P charts, not emagrams. However, the moist adiabatic lapse rate varies significantly based on temperature, and 5.9K/km is well within the range of possibility. The average environmental lapse rate is not the same as the average moist lapse rate because sometimes the atmosphere will be dry adiabatic, sometimes there will be an inversion, etc. But since the atmosphere is often moist adiabatic, the two happen to be similar.