This is probably more about me than the actual game, but when it comes to draft day, I have no option allowing me to pick the player. I can forward to "Next Human Pick" but once there I can't do anything but let the computer pick for me. What am I missing here?

When you click there you go at once to your time selecting the player. Afterward just click on the "war room' and select the player.
Maybe you are missing the "war room" and the cpu selects the player because your time to select is over.