Ok now I remember that piece of logic (sound of brain cells dropping into place like a crazed pinball machine).

Here is how it works for the earth’s rotation. The rate of change caused by the earth’s rotation is maximum when the star rises in the east and minimum when the same star sets in the west. The Doppler is negative throughout this traverse. The Doppler is different when you’re not looking directly east at the start but Trig takes care of the difference.
As the antenna tracks and my computer calculates the chirp needed for the receiver it changes through the day as you would expect and the resulting ‘picture’ of an ET signal at the end of the day is a straight line. Local signals show up as a sort of ‘S’ shaped line tipped on its side (hard to describe this picture in text).

The suns movement around the center of the galaxy is indeed large but its rate of change is not. From one day to the next the Doppler is nearly the same and it takes an entire year for the ‘S’ to be completed.

So it’s the rate of change in the Doppler that is really important not the amount of Doppler. After all we don’t have any idea what the real frequency transmitted by ET is only how we receive it and how it changes.

This was patiently explained to me by Dr. Paul Shuch of the SETI League but it escapes me once in a while.