I'm not convinced they moved the Eugenics Wars. If you look at the language used in "Strange New Worlds," it's very vague; what Pike actually establishes as concrete fact is that a series of conflicts preceded the nuclear attacks which were subsequently known as part of World War III, and that those prior conflicts had multiple names including the Eugenics Wars and the Second American Civil War. That's it.
Meanwhile, PIC "Farewell" establishes that something called Project Khan occurred between 1992 and 1996 -- the same dates TOS "Space Seed" established for Khan's reign on Earth.
To me, this strikes me as strategic ambiguity. To casual viewers or casual fans who don't know every piece of Trek trivia or who might be turned off by the idea of a big divergence from real history in the 1990s, the language implies but does not state outright that the Eugenics Wars might have happened in the mid-21st Century and/or have been the same thing as World War III. To continuity sticklers who don't like the idea of a retcon, the language is vague enough that it does not preclude interpreting a 1992-1996 Eugenics War as eventually leading to a second U.S. Civil War which itself eventually bleeds into World War III.
The language is vague enough that you can interpret the Eugenics Wars as happening whenever you want, really.
In some ways the 1996 date almost aligns with the beginning of the American Cultural Wars. Pat Buchanan declared it in 1992: "There is a religious war going on in our country for the soul of America. It is a cultural war, as critical to the kind of nation we will one day be as was the Cold War itself." The movement led to the "Contract with America" movement and a revamped Republican victory in the midterms in 1994. By 1996 with the reelection of of Bill Clinton, the cycle of stalemate and political conflict began that has not abated in the United States.
1996 saw the ouster of Mobutu in Zaire leading to a series of Congolese civil wars which have never entirely ended and in which so far over five million people have died, to the complete yawning of the rest of the world, and yet it plays out repeatedly as a proxy war for the world's participants.
1996 is also the year that Putin's dacha burned, without him in it, leading to the formation of Ozero and Putin's entrance into public life as an advisor (and eventual public replacement) of Boris Yeltsin.
1996, Ukraine adopted a new constitution, and gave up its nuclear arsenal in the Budapest Memorandum in return for a promise never to be invaded or threatned by the Russian Federation.
Oh yes, we never know what we look at fully until we look backward in retreat and dug in to our miserable present day. 1996 was a pivotal year.
I hope now that a little more light has been shed on Trek's murky 90's through 2070s that it can remain not too far filled in. Picard made clear that even despite the similarities, their 2024 is not going to be ours. It remains a seperate universe, and one still more advanced than the one we live in. We aren't going to launch anyone to Europa in two years, for instance, certainly not from Vandenberg AFB.
I thuoght the Khan reference was to imply that 2024's Soong would focus his efforts on working with augment DNA, leading to the Soong we see in Enterprise.