Doesnt it do that already?

For me the flashearth map has the same center and rotation as the current view when you click the white arrow icon. The zoom is a rough approximation, but is reasonable. (edit: I take that back the zoom is rubbish, will leave that till tomorrow, but the centering does work for me at least)

Its a view based refresh, so as 1 second after you stop, the arrow jumps to the center of the view.

... if not then there a bug along the line somewhere...