Why did Tank Screw Neo in the first place?

The mobile phone used in the Matrix 1 was a Nokia 8110 with a slight modification to make it open the keypad cover automatically.

The agents didn't take over Neo because the movie would have to end there.

They didn't take over him because they couldn't. Just like they can't hit shit. It's all part of the control...

I think that agents cannot take over people that are aware of the Matrix and the existence of them (the agents). It is like a thief trying to get into your house while you can see him through the door window. You will not allow him to enter (block the door, lock it etc.)