Oh, it isn't that hard.
Kennedy lives, and goes on to beat Goldwater in a narrow victory (polls showed Goldwater ahead at times, and let's say Goldwater really did want to beat Kennedy) after winning a series of major debates in cities as was planned IOTL.
The South was Goldwater solid, as was the Midwest, but Kennedy carried Texas, most of the North, and California—all in all enough to win.
As always, the latest civil rights bill gets filibustered in the Senate. The summer riots of 1965 are particularly bad—requiring military intervention in a number of large cities.
Network news carries a number of them live, and the "Silent Majority" is appalled.
The 1966 summer riots are wider spread and last longer than 1965, and the backlash is even worse ITTL, with the Republicans falling only a dozen seats short of capturing the House. Coupled with Southern Democrats, the conservatives have a solid majority.
Kennedy pushes hard on another civil rights bill, having belatedly woken to the fact that he needs to pass this. The House massively weakens it, and irregardless the Senate once again filibusters it.
South Vietnam is not going well, but Kennedy declines to send troops—instead he heavily increases aid and advisors.
With a stroke of his pen, Kennedy begins to bring the sorts of civil rights he can bring in. Essentially he uses the full force of the United States to enforce everything on the books. It's getting nasty in the South, and the black population elsewhere doesn't particularly care that Kennedy is trying. 1967 sees military law enforced in most cities with a large black population.
The riots aren't nearly as loud, under those conditions, but it creates a fairly large gap between between how the black see the situation (police state) and how the rest of the country sees it (required action, because the blacks are going crazy despite Kennedy helping them).
Early 1968 finally sees a civil rights bill of some force passed, but as happened IOTL this actually results in worse black riots…*leading to a wholesale rejection of the civil rights agenda by the white middle class population.
Distracted by his illness and growing rumblings into his varied and wild infidelity Kennedy is not on top of things. Johnson declines to run in late 1967, citing health concerns and so the 1968 Democratic Presidential race boils down to George Wallace, a reluctant Hubert Humphrey, and some others—it doesn't matter because Wallace, supported by the South and doing quite well in primaries will be the nominee. Let's say Wallace takes a high-profile Northern moderate (not a liberal)….
Meanwhile the Republican side is Rockefeller (who, butterflies, didn't run in 1964), Reagan, and Nixon. Nixon is squeezed out between Rockefeller and Reagan running open campaigns from the beginning of the primary season instead of back and forth (Rockefeller) or not declaring until the convention (Reagan). At the convention Rockefeller cuts a deal with Thurmond who fatally weakens Reagan's southern support in return for a conservative VP candidate.
(Can you tell I'm feeling too lazy to look up the right VP choices?)
A classic Southern Democrat versus Liberal Eastern Republican face-off ensues, with Wallace capturing a slew of Northern industrial states and carrying the South—making a Rockefeller electoral victory impossible.
President George Wallace backed by a conservative Congress stops enforcing civil rights, and begins a large-scale crack down. Naturally this brings about huge black riots, and the police step it up. By the time they're put down, for now, thousands are dead and across the US most major cities are on the brink of utter chaos.
Meanwhile President Wallace is determined to save the faltering South Vietnam, and pours troops in.
The 1969 summer riots are nuts. Having prepped for several months following their previous defeat the radical black leadership strikes hard and fast, seizing key city infrastructure and mounting effective road blocks along major arteries.
Black military troops refuse to intervene at home and simply stop fighting in Viet Nam, tying up limited American military resources. The draft lottery is expanded, and the college exemptions dropped as white liberal colleges join black revolters.
Although generally put down, the 1969 summer riots continue to simmer in guerilla action. Widely speaking the white middle class is behind Wallace, seeing the blacks as tossing away the civil rights they were getting because of (as they see it) some minor required peacekeeping. On the other hand other minorities and the poor are being swept up these riots, and white liberals are firmly on the side of the blacks.
The 1970 midterms see a mix of conservative Democrats and Republicans coming into office, and the liberal Democrats are more and more horrified by their fellow elected officials. Following the 1970 results a mix of liberal (mostly) Northern Republicans and liberal Democrats form the Progressive Party (a name both sides can agree on) and begin to prepare a '72 Presidential run. Robert F. Kennedy is perhaps their most prominent elected member (let's say JFK encouraged him to run in New York as RFK did IOTL after JFK's assassination), with a sorrowful John F. Kennedy announcing that he cannot support his own Democratic Party.
Meanwhile the combination of Viet Nam pulling troops overseas and increasing violence spreading out of the cities (particularly the Southern countryside, with a large number or rural blacks) has expanded policing powers in a broad and generally deadly fashion.
In the Northern cities a few have been somewhat peaceful, but they're mostly marked by the targeting of elected officials—Mayor Lindsay of New York was assassinated in the summer of 1968, and Mayor Daley of Chigaco now walks with a pronounced limp and a very large police escort.
In response even liberal or sympathetic to black local public figures have come down hard—Governor Agnew of Maryland basically bringing in full scale segregation.
Thus the 1970 situation sees the US on the verge of an actual, mostly racial, civil war. There are a lot more police, and hence a lot more incompetent, corrupt, and brutal police (very low standards as they are competing against a larger than OTL draft as well as an even greater than OTL need for police).
The white population will do anything for order, and they believe civil rights is an unacceptable option given that the really nasty black riots started with the passing of a powerful civil rights bill (yes and no, if Wallace had not been elected blacks would have quieted down the following year).
The 1972 election is shaping up to be an odd contest: Northern states look likely to go Progressive, Midwest/Southwest states Republican, and the whole South will be solidly Democratic; California and a few other states are up in the air because of the three-way competition. That leaves no one with the electoral college required but the House is solidly conservative and quite Democratic—Wallace will win an election thrown to the House.
The Republican Party is torn. Wallace has embraced the old Southern Democrat/conservative tradition of pre-New Deal politics while the Progressives are basically the old Teddy Roosevelt Progressive. The Republicans, then, are mostly the Taft Republicans of 1912 and are uneasy both with Wallace and the Progressives: Wallace bothers them more. Quiet talks have gone on between the Republicans and the Progressives about a joint ticket, but Reagan and RFK (their respective leading lights) are each unacceptable to the other side.
Meanwhile the violence gets worse. From Viet Nam the black population is getting a lot of weapons, supplies, and drugs to finance their operations which is just making the situation that much worse.
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That's as far as I go for now (and it's already pretty nasty). I don't think it's terribly realistic, but a lot of people forget how bad it was getting in the middle-late '60s. I don't consider this ASB (some details, maybe) or out of the question. It mostly rests on radicalizing the US population more than OTL, and having a larger white backlash coupled with delayed civil rights and hence a larger black backlash.
President Wallace is the easiest figure to work with, but Nixon would almost do given somewhat different conditions.
The Democratic Party practically was two parties, once FDR came to power, and I don't consider the split unreasonable.
I honestly do think Kennedy would have meant things had a chance at turning out better for the US… but I'm quite certain he also means a chance for much much worse—the Johnson legislation program rested on a mix of Kennedy's death & the '64 landslide victory…*take away both and civil rights aren't getting passed in any strong form: especially with an even larger Republican swing than OTL in 1966.