Time remapping does not change the timing until you add, then move or change the value of the keyframe. It always puts a keyframe at the head and tail of the footage layer. If you want to add additional keyframes nothing changes until you move the keyframes or change the value. I'm not sure what you are trying to do, but sliding the layer back and forth isn't going to change the speed of the playback. Time remapping also does not change the in and out point of a footage layer.
If you want the start and end keyframes of a time remapped layer to always be at the in and out point you have to pre-compose the layer first. Then, if you want the same in point in relation to the comp time but you want the video to start earlier you are out of luck because there are no earlier or later frames in the pre-comp to move around.
I think you need to type time remapping in the search help field in the top right corner of AE and do some studying. Nothing in your video explained what you were trying to do with the speed of the video.