Hm, Plutarch himself wouldn't be a good source for the mac language, given he wrote many centuries after even the era of the peloponnesian war. Mac did play some role in the peloponnesian war (iirc Amyntas was king then, so after Alex I), yet it still wasn't that considerable a force to be recognized as greek due to just helping as second-rate power to avert some athenian landings/campaigns. Furthermore, as always, it wasn't unique in this in power-balance; you certainly don't see Athens recognizing actual barbarian tribes as greeks just because they were in the habit of going to war against Macedonia (recall the lie spread by Demosthenes, that Alex fell in some ambush by those barbaric neighbours). I think it would take some non-tenuous tie to have greek states identify Macedon as part of the greek world, even in partial standing (it even got to have athletes before its rise to any importance, in the olympic games, which also is not just symbolic). Later on it got to be part of the holy war, which again wasn't for any non-greek state (the way it got to be part of it was convoluted; Thebes did leave for false reason, and no one expected that Phokis would actually melt the statues at Delphi to pay for a large mercenery force; still, Mac was hardly the only power able to defeat those mercs (it even almost lost to them).
Re Mac being a kingdom: yes, it was. Kingdoms had been also in the greek world in the mycenaean era, and moreover something very similar to a king existed in the classical era too in greek states: a tyrant. Tyrant just meant someone who rose to power without legitimate means (eg in a coup), yet if he got to power he was in a position very similar (if not the same) to a king, so arguing that Mac being a kingdom shows it isn't greek... isn't really by itself something making the argument stand taller, imo.
Also, most politiai were controlling an area far larger than an actual city (which Polis means), so territory-wise they certainly were often larger than any peripheral kingdoms, as those in the north or east of Thessaly. Speaking of Thessaly, that is home to Achilles, and the city of Iolcos, which again is one of the most central locations in the greek myths (argonauts etc). It is to be expected that stuff differ in other greek areas than the peloponnese-attica-thebes one. Eg Thessaly had considerable use of cavalry early on, but wasn't unique in that in the greek world; asian minor greek colonies also did, eg Kolophon was even (supposedly) named after its famous cavalry.
Last but not least, re language: Macedonia itself is a greek name, apparently tied to the term "makednos", which afaik means "tall". Greek state names do tie to such, somewhat often (locroi being another example, for Locris). Other names tend to derive (arguably) from the priestesses of each state's acropolis (the Athenae etc)