A quick correction:
I technically misspoke earlier in referring to Garithos’ command as one over an Alliance army. We think of it that way because the Army of Lordaeron had elvish and dwarven troops among them in-game (based on WC2 and 3/4s of the original member races in WoW).
According to lore, though, that wasn’t so much the case. Garithos did not actually control an Alliance army (i.e., one in any way co-directed by the Alliance heads of state), per se, but merely a part of the Army of Lordaeron (a single human city-state). Kaelthas bound his forces to that army in memory of the old alliances they’d had (with Arathor, with the Alliance), but he was technically serving under just the Army of Lordaeron.
However, that was the particular army, rather than technically the Alliance, to whom Kael’s forces gave their allegiance, as it was the only group that actually had forces and supplies near Quel’thalas when they were needed. Remember, there’s a reason the WC3 Campaign calls the faction Human, not Alliance. It’s the story of Lordaeron, the northernmost human nation, even if it carries over iconic mixed-composition forces from WC2.
Neither Varion or the Stormwind leadership nor Magni or his councilors at Ironforge would have heard a damn thing about what Garithos was doing in time to make their own commanders intervene. News, like those forces, would take months to travel except by gryphon, in which case it’d still take over a week. They can only send their troops and hope for the best. And what Garithos was doing would only have been at all visible to those troops for less than three days.
So no, a single citystate’s military head doing something horrible does not mean that Ironforge and Stormwind leadership were necessarily complicit in that. They’d have no way of giving orders beyond sending their troops and commanders over.
You may as well, again, accuse every member of the Horde of being complicit in the mana-bombing of Theramore. Were they aware of Garrosh’s plan? Very few. But hey, let’s assume that all related factions, even those without any avenues of communication or oversight, are complicit, right?
The Alliance, which was a council of heads of state between Stormwind, Khaz Modan (Ironforge), and only tenuously Lordaeron even before its head of state was murdered and no one was able to replace him, had no say over what happened to the Quel’dorei. They did not put Garithos in charge. The only thing that put Garithos in charge was every other officer in the eastern region having been killed as the Scourge pushed towards Quel’thalas.
Garithos, being only aligned with Lordaeron (not Stromgarde, not Stormwind, and certainly not the broader Alliance) could only be ordered about by the Alliance if the compact between Terenas, Varian, and Magni could force Terenas to do so. Terenas couldn’t, though, because he was dead. Meanwhile Varian and Magni were on the southern continent. Alliance in this case is a bit of a misnomer, more a vague idea of historical reference than technically a chain of command, so no, Garithos’s actions are not a sign of approval from Alliance leadership higher up.