No ET! Ever?

[Anomalocaris]

That picture you show reflects the precession of equinox.   It has nothing to do with earth's motion or proper motions of target stars and therefore does nothing to increase the difficulties of targeted transmission.

Your original claim was, "the stars are in pretty much the same positions as when they were observed by the Greeks / Ancient Chinese." (emphasis mine).

In any case, the proper motion of the Sun around the center of the Galaxy (at nearly 700 times the speed of sound), would be sufficient to cause a narrow band radio transmission to miss the Earth, not to mention the Earth's motion around the Sun (around 60 times the speed of sound).

I suspect the diameter of the earth's orbit around the sun would be way smaller than the dispersion diameter of radio or laser signal pulse or beam sent to us from the likely distance of any technologically comparable civilization.  So earth's motion around the sun doesn't matter.  Either much of the solar system will be illuminated by the signal, or it won't be.  Conversely the motion of the earth around the sun also won't make our transmission miss the target for the same reason.

The proper motion of stars caused by our and the star's independent orbit around the Milky Way isn't a real tough problem.  We can detect the proper motion of some near by star being caused by the small pull of  planets.   We can also measure the radial motion of the star to an even higher degree of precision, thanks the efforts put into exo-planet searches over the last 25 years or so. So we can define the motion of nearby stars in 3 dimensions to extremely high degree of accuracy and precision.

The main problem is the measuremeant of the absolute distance to the star.  We don't have the ability to measure stellar distances to a comparable degree of accuracy.   So we can't tell with anything close to the same degree of precision how long it will take exactly for a signal to reach the target star.  Hence unless we blanket the sky with signals, it would be very tough for us to send any short pulse signals in such a way that it would arrive at the same place and time as the target.