I was going to post this on the Sicilian thread, but seems better suited here . . . .
Also, you have to understand that the Drews "theory" isn't saying that the Mycenaeans don't have steppe; it's saying the "route" was from the steppe, through the Caucasus and across Anatolia.
It doesn't matter which route was taken. The autosomal signature of the Mycenaeans remains the same: they had very little steppe.
I completely agree with you that their culture is too sophisticated and advanced to only have steppe influence, but don't forget that they also have Iran Neo like ancestry, as do the Minoans, who had a sophisticated civilization before them, which came from influence from Anatolia. They adopted a lot of their culture from the Minoans, and who knows, perhaps they got the Iran Neo like genes in part from admixture with them.
So, yes, Lazaridis found STEPPE in the Mycenaeans, although not in the Minoans, but he didn't come to a conclusion as to whether the route was from the steppe west and down through the Balkans, which I still think is more likely although I'm not married to the idea, or from the steppe through the Caucasus, and across Anatolia to the Aegean, which would be the Drews' Greeks from the east hypothesis.
Drews locates the PIE homeland not on the Pontic Steppe, but "south of the Caucasus, somewhere in the area now covered by northeastern Turkey, the northwestern tip of Iran, and the Soviet republics of Azerbaijan, Georgia and Armenia." I am quoting from Drews' 1988 book The Coming of the Greeks, page 148. Drews' argument is narrowly focused on chariot technology, and it was in Armenia that the trees were to be found for building spoked wheels, as well as the craft-sophistication in working with wood. (I haven't read Militarism and the Indo-Europeanizing of Europe, so not sure if, or how, his views have changed.)
Drews distances his argument, in TCotG, from the "racist" views held by earlier historians: "Initially, racist presuppositions impeded some historians from assigning much importance to the chariot in these movements. Believing in the existence of an 'Aryan' race, and in the superiority of that race, they were not inclined to attribute the Indo-Europeans' success to a gadget (even Gertrud Hermes allowed that the Indo-Europeans' rise to power was rooted in their moral qualities as well as in their chariots). The reaction to Aryanism was equally uncongenial to the idea that the PIE speakers began as charioteering conquerors. In such a picture, even if one gave it only a superficial glance, one was embarrassed to see that the PIE speakers still looked like a Herrenvolk, lording it over Semites, Hurrians, and the pre-Indo-European inhabitants of Europe. Another school of thought was not averse to seeing the chariot as one of several means by which the PIE speakers prevailed, but was not inclined to believe that mastery of chariot warfare had motivated them to embark on their expeditions" (page 147).
Although I would not call him a racial supremacist, Ricardo Duchesne's thesis of the "aristocratic egalitarianism" of the Yamnaya as the primordial ground of (virtually all) later European greatness is reminiscent of earlier theories in its emphasis on the (supposedly) unique moral & genetic qualities selected for on the Steppe. Homer & Plato, in Duchesne's treatment, are exemplars of the unique spirit of the Steppe, of the competitive drive of aristocrats to distinguish themselves from their peers, to win fame, status, recognition. Plato-to-Caesar-to-Galileo-to-Goethe is the agon of the Steppe sublimated. It is as if the other components of European culture and genetics were of no consequence.
Both Duchesne and Drews posit an "elite takeover" of the Greek mainland, followed by Crete and other islands (though Duchesne is a speculative historian, at some remove from primary research). According to Drews, "We may still suppose that when PIE speakers took over the best parts of the Greek mainland--a land that seems to have had almost no political organization, and a populace that seems to have set little store by military prowess--they controlled an alien population perhaps ten times as large as their own. Such was the ratio of helots to Spartiates as a much later time, and even the Greeks who took over a relatively well-organized Knossos ca. 1450 B.C. may not have numbered over a tenth of their Minoan subjects. One can only guess at the numbers involved in 'the coming of the Greeks,' and I shall hazard mine: if at the beginning of the sixteenth century B.C. the population of the Thessalian plain, the Argolid, the Eurotas Valley, Messenia, Boeotia, and Attica was approximately three-quarters of a million men, women, and children, the PIE-speaking conquerors may have numbered no more than about 75,000 men, women, and children" (page 195). And Duchesne's gloss on Drews: "While the Mycenaean minority 'did not ethnically transform the land,' it superimposed its language and culture, and thus it 'Indo-Europeanised Greece'" (Uniqueness of the West, page 366).
The Lazaridis 2017 study is not, at first blush, inconsistent with the theory of an elite takeover. Mycenaeans could owe 75% of their ancestry to Neolithic Anatolia, but still have their high culture decisively shaped by an alien elite. The only finding of Lazaridis that undercuts this theory is that the Mycenaean buried in an elite grave did not differ genetically from Mycenaeans buried in common graves. But I think Lazaridis looked at a total of four graves?
The Drews thesis is radical & heterodox. He emphasizes circa 1600 BC as the key period of military conquest, not only in Greece, but in the Carpathian basin, and then points westward. In general, he adheres to the Gamkrelidze-Ivanov thesis, whereby it was only the "European" branch of the Indo-European family (including the Celtic, Italic, Germanic, Baltic and Slavic) that lived on the Pontic Steppe. The Hittite-Indo-Iranian-Greek-Armenian branch followed a different course. Drews' chronology at odds with prevailing scholarship. He imagines a very rapid pace of language change pre-Homer, and he evinces little interest in "satem vs centum."
My own intuition -- which is in no way history or science, just a man's intuition! -- is that Drews is right about the PIE homeland and the takeover of the Greek mainland circa 1600 BC by Armenian warriors (who sailed on ships from Armenia to the Aegean, with horses and chariots on board). But the Armenian takeover of Greece was preceded by some 5,000 years of CHG and Iran Neolithic slowly making its way, sometimes in trickles, sometimes in waves, across Anatolia, into the Aegean, mainland Greece, Southern Italy, Sicily. The taking of Greece in one fell-swoop was merely the most dramatic chapter.
And the Armenians who became Mycenaeans carried perhaps 20% Steppe blood (don't have exact percentage ready-to-hand), in addition to CHG/Iran and Anatolian. But they were not fundamentally shaped by the experience of the Steppe; their culture, which would later spread throughout Magna Graecia and the Classical World, was not a product of the Steppe. It originated in Armenia. The road taken means everything.
Such is the "sports team" that I support. As a half-Calabrese, I tend to over-identify with Minoan-like peoples. I am also part Irish & English, but don't identify with Steppe ancestry.