I have seen the argument most forcefully put forward in Christian Meier's From Athens to Auschwitz wherein he argues that what we think of as "Western Civilisation" started with democratic Athens and ended with the Holocaust. His main reasoning being a common culture based on free debate and other, in my opinion, way too positive stereotypes, which he claims led to a "European Miracle" around 1500 and from there to modern society's greatest achievements and horrors - where he conveniently chooses to end one era and start a new one.

I'll gladly give you some more details on his reasoning, but I honestly don't think it's worth the time of either one of us. What he writes is not untrue, but to make the connection so direct and all-embracing is just, in my humble opinion, rather silly. For example, he places a lot of importance on the Greek writings that came through Italy from Byzantium in the Rennaisance, but mentions nowhere the huge amounts of additions (scholia) made by Byzantine scholars over the centuries, even though they would have carried almost as much weight as the original text.

I'm by no means an expert, but unless someone far more convincing than Meier that I don't know of is arguing for this idea, I think we can mostly say that it's, if not wrong, then grossly oversimplified.

I think your question introduces a self-affirming vicious circle. Ancient Greece wasn't necessarily distinctly or uniquely "Western" compared to earlier or other civilizations, it mostly inspired and nurtured qualities we consider to be the basis of our "Western Civilization", so we impose a "Western" identity upon them.

However, we could argue about terminology or historical discourse all day, but let's take a look at popular opinion for a moment. Why is Ancient Greece often seen as the birthplace of "Western Civilization"? In order to answer that question, I think we need to look at our preoccupation with scientific advancements in historical discourse. Throughout historiography, there has been a lot of focus on scientific advancements as a measure of "progress" and the road towards "modernization". Though this perspective has been criticised and even refuted by most historians, it's still very much alive in popular thought and even history education on a non-academic level.

Considering how we perceive scientific advancements as the most defining aspect of "Western Civilization", we have to take note of the profound impact some of the scientists, artists and philosophers of Ancient Greece have had on "Western Civilization". The Roman elite was just as obsessed with Greek civilization as most scientists in later times. There isn't a single period in "Western civilization" where Socrates, Plato, Aristoteles, Archimedes, Pythagoras -the list goes on- weren't considered to be extremely relevant to their specific fields. This legacy was kept afloat thanks to proximity and a conscious effort to preserve their works.

Because it is. Europe was once the name of a small geographical region north-west of the Bosporus. Greeks named it after the river, crossing this region, (Euros) Hebros; The region that was located just across the straights, on the opposite bank, was called by the greeks Asia (not the whole continent, just the small region on the opposite bank of dardanelles). So it was not before the trojan war when the meaning of these two areas become symbolic - or, to be more precise, not before Homer configured his epic poetry about the trojan war - this is when the dichotomy rose - it was greek versus troyans, but since they lived on the opposite banks of dardanelles, they also became europeans versus asians. For Homer the europeans became "us" and asians became "them", it was not about greeks and trojans anymore - the trojans (perhaps a hellenic tribe themselves) became the archetypal "others" - different by all means and in a despicable way. In the pick of the hellenic antiquity Europe rose to a continent - Asia as well. The West copied this homeric notion during the renaissance - this was the geography of the awe inspiring greeks at the end - and became Europe - naturally, self identifying with the greeks. Source - Edward Said, "Orientalism". Other sources as well.

I can't speak to other reasons, but during the enlightenment there was a (racist) push to put Greece as the beginning of Philosophy. The idea being either that they came up with it whole cloth, or they were so exceptional that they were essentially the start. You do have many great thinkers coming from Greece (Thales, Pythagoras, Socrates, Plato, Aristotle, etc.), and coupled with the fact that they are European made them a good focal point for racist re-imaginings of history that exclude ancient older civilisations.

And slightly more speculatively, I imagine that the Hellenisation of Rome, and the subsequent domination of the Roman empire had something to do with it.

Source for the first part: Africa, Asia and the History of Philosophy by Peter K.J. Park