Whether we call classical Greece "western" is not is ultimately not all that relevant. What matters more to me is that, reading the ancient Greek sources, we see attitudes, philosophies, dreams, and goals that are immediately recognizable as being "us" in an important way.

Consider the justly famous and beautiful funeral oration of Pericles:

"Our constitution does not copy the laws of neighbouring states; we are rather a pattern to others than imitators ourselves. Its administration favours the many instead of the few; this is why it is called a democracy. If we look to the laws, they afford equal justice to all in their private differences; if no social standing, advancement in public life falls to reputation for capacity, class considerations not being allowed to interfere with merit; nor again does poverty bar the way, if a man is able to serve the state, he is not hindered by the obscurity of his condition. The freedom which we enjoy in our government extends also to our ordinary life. There, far from exercising a jealous surveillance over each other, we do not feel called upon to be angry with our neighbour for doing what he likes, or even to indulge in those injurious looks which cannot fail to be offensive, although they inflict no positive penalty. But all this ease in our private relations does not make us lawless as citizens. Against this fear is our chief safeguard, teaching us to obey the magistrates and the laws, particularly such as regard the protection of the injured, whether they are actually on the statute book, or belong to that code which, although unwritten, yet cannot be broken without acknowledged disgrace."

""In short, I say that as a city we are the school of Hellas, while I doubt if the world can produce a man who, where he has only himself to depend upon, is equal to so many emergencies, and graced by so happy a versatility, as the Athenian. And that this is no mere boast thrown out for the occasion, but plain matter of fact, the power of the state acquired by these habits proves. For Athens alone of her contemporaries is found when tested to be greater than her reputation, and alone gives no occasion to her assailants to blush at the antagonist by whom they have been worsted, or to her subjects to question her title by merit to rule. Rather, the admiration of the present and succeeding ages will be ours, since we have not left our power without witness, but have shown it by mighty proofs; and far from needing a Homer for our panegyrist, or other of his craft whose verses might charm for the moment only for the impression which they gave to melt at the touch of fact, we have forced every sea and land to be the highway of our daring, and everywhere, whether for evil or for good, have left imperishable monuments behind us. Such is the Athens for which these men, in the assertion of their resolve not to lose her, nobly fought and died; and well may every one of their survivors be ready to suffer in her cause."

https://sourcebooks.fordham.edu/ancient/pericles-funeralspee...

These are words that you can imagine coming out of the mouth of a figure like Lincoln or FDR. Pericles reaches into many of the same ideals we still venerate today. I've never seen anything chiseled into some Egyptian obelisk or scrawled onto a Sumerian tablet that feels so, well, familiar, so recognizable, so modern.