Maybe Noonien-Singh is a common place name on Earth, like Smith. Which is why Kirk couldn’t connect the dots.
This tracks...
Remember that in "Space Seed", Khan was NOT treated as like, Space Hitler. He was treated closer to like a Napoleon. Kirk calls him something like the "best of the tyrants" or some such and has a kind of admiration for him.
Noonien-Singh may well be a common name, people having renamed themselves in adoration of Khan. It's taken on something of a life of its own after the episode, but going by straight TOS... historical Khan wasn't regarded as "evil", merely a conqueror.
The New United Nations is a thing in 2036, so maybe under that banner.
Is also possible that Archer’s great-grandfather served during the overlap of the Gulf War and the '90s Eugenics War, and had a first encounter with the Augments. And then returned to service, or even led troops, in the first decade of the 21st century Eugenics War.
The United States would still exist at that time...
I do like you splitting up Eugenics Wars. In my own headcanon history, I split them into two conflicts that are collectively known as the Eugenics Wars. I have a timeline of 1st Eugenics War (called something else at the time) in the 90's, the Second American Civil War in the 2030's (SNW implies they are connected... I take it as a new attempt at an augment to seize control), 2nd Eugenics War in the 2040's (spiraling out from the 2nd ACW to engulf the world), that ultimately gives way to outright World War 3.
On a tangent in a compacted tl;dr format, I kind of disregard SNW's "Tomorrow and Tomorrow and Tomorrow", but have Khan show up in the 90's and a bunch of augments seize control mostly in Asia, and promptly go to warring against each other. They are eventually defeated, leaving a power vacuum of these collapsed empires. Augments don't completely disappear, but the science is tightly regulated. The 2nd ACW is when an unashamed augment makes a power grab at the US Presidency, to surprisingly wide support. This augment ultimately loses the election, but his supporters are wildly loyal and reject the election results. The US erupts into civil war, and a few new augment empires rise from the ashes of the former augment empires, once again controlled by either first generation augments or their offspring. The augment rebellion is put down in the US, ending the 2nd ACW but now the new empires used the destabilization of the US for a new wave of conquest. Much of the world bands together and they are taken out of power once again, however the new leaders of those empires do not just want to sit and let the other nations roll over them. The Eastern Coalition forms from the remnants of the second augment empires, and the conflict evolves into World War 3.