Speed is not a function of how fast you can move, but of how quickly you can close distance. As such, speed is not a matter of agility, but of directness.
For example, if I can close the distance between myself and the opponent directly in a linear fashion, I can technically reach the opponent before they reach me. This is because if you can close distance with your whole body, it works as a force multiplier. Not only are you closing distance with one body part (the way an arm would be in a punch), but with multiple body parts. Your lower half is closing distance, your waist is moving closer, AND your arm is reaching out. All these movements together allow for a speed that is quicker than any individual slow or fast twitch muscles on their own. Of course, having quick movements is also of benefit and can work well in conjunction with the rest of the body moving forward.