In one of my own high school history courses, I distinctly remember my teacher mentioning that the Greek city states and cultures did not consider the Macedonians to be Greek, otherizing them as backward off-brand cousins or something to that effect. I am of course referring to [this Macedonia](https://en.wikipedia.org/wiki/Macedonia_(ancient_kingdom) and not the Republic of North Macedonia, which is of a Slavic, Balkan origin. Regardless, Alexander the Great and his generals were famed for spreading Greek culture across the ancient east, from the Adriatic Sea to the Hindu Kush valley. The names of Alexander and his generals are all clearly Greek, etymologically speaking, and the same was true for Alexander's father. His mother, Olympias, was from Epirus, and the Argead dynasty was of Dorian origin.
I can't help but feel that this is an all too common misconception, like the idea that all medieval European people only ever bathed once a year. But is there anything from period texts of the ancient world written by those southern Greek cultures and states which suggests that those from Μακεδονία weren't considered to be properly Greek?
Yes, there is plenty of evidence that the Greeks considered the Macedonians barbarians (that is, non-Greeks) when it suited them. This does not mean that the Macedonians categorically were not Greek. It does mean, though, that their Greekness was apparently open to question, and that authors with different agendas could exploit this at their convenience. It is not possible for modern historians to dispense with the ambiguity and make a sovereign declaration one way or the other. In this older thread I explained why the Greekness of the Macedonians was never a simple yes-or-no question.
To respond to your specific points, firstly, the Greek-sounding names of the Macedonian elite don't tell us all that much. While there are now a few scraps of written evidence from the mid-fourth century to suggest that some people in Macedon spoke a Dorian dialect of Greek, all this confirms is that the elite of Macedonian society seems to have been hellenised in this narrow sense (though for instance their burials were not like those of the Greeks further south). Greek identity was what the royal family claimed for itself anyway, and some ancient authors (including Herodotos) were prepared to accept their claim. But it does not tell us whether the Macedonians in general spoke Greek or counted as Greek. Sources on Alexander pointedly have him speaking Macedonian when he needed to stress his bond with his soldiers; it is not clear whether this is a reference to the aforementioned thick dialect or whether mass and elite literally spoke different languages (which is certainly not unheard of in history).
Secondly, the fact that Alexander's successors were famous for spreading Greek settlements across the Near East (whether they also spread Greek culture is doubtful) does not tell us much about their own ethnic identity. The communities they created were built on the model of Athens, not Pella. They did not replicate Macedonian hierarchical customs or political organisation, but borrowed those of urbanised and democratic city-states. The reason is obvious - there were many more Greeks in the world than Macedonians, and new settlements needed a ready population of settlers. As for the generals themselves, the sources are pretty careful to distinguish between Macedonians and Greeks: a general like Eumenes of Kardia (a Greek) could never hope to reach the same status as Macedonians like Antigonos and Seleukos in the eyes of Alexander's Macedonian soldiers. For them, it really did seem to matter that they were specifically Macedonian, even if that was just a subset of the Greeks.
In short, the inclusion of Macedonians among the Greeks was at best uneasy. There were some on both sides who wished to promote it, while many (also on both sides) actively rejected it. The motives for doing both were always political, and as always with such contested questions, there are no easy answers.