Well of course the acceleration is greatest short before contact. however that doesn't mean players first swing slowly and then accelerate gradually.
the reason for the late and steep incline of the velocity curve is that the final stages of the kinematic chain fire quite late in the swing. and just like a whip because those links are the lightest in the chain they also create the most speed.
earlier in the swing you accelerate just as hard, however at this point the slower and bigger (but stronger) parts of the body are doing the job. in that phase not much speed is generated but a lot of the energy needed for the final parts of the chain.