Spooky action at a distance, as we presently define it, requires two entangled particles. So you're back to the light speed problem since in order to entangle particles they need to be close together. Then you have to send half of the entangled particles to the distant place and as they are ordinary matter light speed will apply. All before you can commence FTL communications.
As for alien civilisations being out there long before we were... most of the comments assume that once a civilisation is established, it does not die out or switch to something more efficient than radio. What you are actually hoping for is that the earth intersects a circular shell of communications that is loud enough to stand up above background noise at the frequency you are investigating. Many frequencies are useless for interstellar comms, but that still leaves an awful lot to search and some may think it wise not to attract attention by shouting across space!
