I’m not convinced that alignments would help. In Greece itself the alignments are not so strongly associated with astronomy. In many cases temples are built over older sites, so follow the older alignments, which might just be topographical instead of astronomical.
If I remember correctly Retallack published in Antiquity showing that Greek temple dedications in Greece were more associated with local geology or flora than astronomy. Sadly, it doesn’t transfer to Sicily where I was looking.
There’s also another paper by Bowen on a change in Greek Astronomy. He notices that there’s a major shift in what the Greeks are interested in around the Hellenistic period. I think this is connected to a shift in politics.
People commonly say ‘As above, so below’ and argue that civil society is ordered after how people view the cosmos. It could work the other way round. So Epicurus, the radical democrat, sees a cosmos working through atomic theory. Aristotle in contrast sees a universe with a clearly ordered hierarchy, around the same time the Macedonians are setting up their empires.
I think the Greeks could have discovered the precession of the equinoxes earlier, but I don’t think there was the interest in doing that. When we’re talking about Greeks we’re talking about anything between 800 BC or earlier to 146 BC (or later). I think it’s likely that the paucity of sources before thr 4th Century BC hides the differences between the earlier and later Greeks.