I don't dispute it had massive hype at the time. I wasn't even an xbox fanboy but I still played it with friends who had the system. I still think its long-term cultural impact is rather minimal, the biggest influences of the series on gaming as a whole came from Halo 1 and 2. And as a sibling comment notes, Call of Duty Modern Warfare released only a few months later and was similarly massive, but unlike Halo, CoD has kept up its appeal and impact over the years, from MW2 and on. (Even newer Halo games ape CoD more than their own roots.) Most gamers haven't played Halo, though hopefully most would recognize its title. In comparison, Minecraft (which came later), has such broader name recognition as well as more people who have actually played it, and continues to have strong and popular cultural impact to this day. It didn't have a flashy launch, though, sure.
Perhaps I'm just missing your argument. Is your argument mostly about the immediate "pop" surrounding the launch, rather than any sort of longer term cultural impact? If so, then sure, I'll grant Halo 3 had one of and perhaps even the largest launch hype of all time from the physical release age. Still it's not like Halo 2 wasn't hugely popular at its launch too, though 3 was probably bigger. And of course the already mentioned Modern Warfare had a big launch, though by MW2 the whole culture of launch hype with midnight releases and camping out and so on was just about dead thanks to digital distribution and such. You had other games later on with crazy marketing too -- how are super bowl ads relevant? Dante's Inferno (2010) is a game I've never played but it also had a super bowl ad I've never seen until now. I'd dare say its impact is far less than Halo's, both at its launch (seems to be a God of War clone) and since.
But if you really are arguing that nothing has had a bigger long-term cultural impact since, the single counterexample of Minecraft is enough to stand against that. Speaking of God of War, there's another franchise that has probably by now out-shined the Halo series for cultural impact. (And maybe worth arguing that it probably contributed to Halo 4's dumb "Press button to beat the Didact" QTE?) I'd even put 2011's Dark Souls higher for birthing a new subgenre term, the "Soulslike", even if it took until Elden Ring for FromSoft themselves to hit the pinnacles of financial success and launch success with the formula. And so many other games.
>MW2 the whole culture of launch hype with midnight releases and camping out and so on was just about dead thanks to digital distribution and such
Not quite. MW2 was the largest video game launch on record by pre-orders, revenue and units sold when it was released. Its release broke nearly all of Halo 3's release records. There was no digital release at launch. It also had a massive advertising campaign. There were large ads in every store with an electronics or games sections to an extent that I haven't seen since.