That depends. One thing to remember is that until September, Nixon led Humphrey by about 15 points. It was thanks to the labor unions and LBJ going into DEFCON mode that many Northern Wallace types came back into the fold, and then the SLC speech won acceptance from the anti-war movement. So many people outside of the Deep South who are still typically thought of as "Southern Strategy" voters ended up voting for Humphrey.

The "Southern" Strategy, in the conventional sense of specifically going for what is thought of as the South? Yes, he would have won if we assume that Nixon won everything that he did in 1960. The South changing to Republicanism was a long process that only really completed during the late Clinton years. It reflected long standing changes in the New Deal coalition over the course of decades, not one or two elections. I personally think the "Southern" part of the strategy shows more when it comes to what Nixon did in office, particularly with the Supreme Court and with busing, than him getting into office.

Let's do the electoral math in the former Confederacy, plus Kentucky for good measure: Wallace won the bulk of the Deep South, in Louisiana, Alabama, Arkansas, Mississippi, and Georgia. Don't see that changing, regardless of what Nixon does, at least not in 1968-far too soon after the CR Act(which Nixon, unlike Reagan or Bush, endorsed), and Nixon isn't going to sacrifice 20 states for 5, 5 states that had nowhere near the electoral impact that more traditional GOP leaning states like California or Ohio did. They'd only go for Nixon if Wallace disappears from the equation, which was what happened in 1972. Humphrey won Texas thanks to LBJ, who was still extremely powerful in Texas. That's another non-starter for Nixon unless Johnson disappears or prefers Nixon to whoever the Democrat nominee is. (Which is one reason I don't think RFK would have beaten Nixon, contrary to Boomer legend. LBJ would have sooner had Alexei Kosygin as his successor, and Texas was 25 electoral votes. Since the Democrats already had New York and Michigan, RFK could have only made up for that by winning California-Nixon's home state that elected Reagan as governor in a landslide in 1966 based off of a tamer version of feelings that would exist in 1968, and a state that had the most formidable GOP machine in the nation until 1992, so highly unlikely, no matter how much RFK would have appealed in inner city LA-or Illinois AND Ohio, which would have been a stretch given Kennedy's alienation from the Johnsonian bosses and the race riots that had taken place in major cities in both states. Winning just one would lead to a Humphrey style loss. RFK also faces more-serious-than-usual image problems in Illinois, arguably the state that had the worst disorder of all in 1968. The hippies were for Gene, not Bobby, but the typical Nixon voter wouldn't have cared. Even Daley wasn't God, as the 1968 election showed. Not completely impossible if literally everything goes right for RFK, given his ability to appeal to the Wallace voter and the black man at the same time, but not likely in the slightest.)

So that leaves TN, KT, VA, NC, SC, and FL. Nixon had already had won four out of the six-Florida, Virginia, Tennessee, and Kentucky-over Kennedy in 1960 thanks to his overall conservatism. All of those states had went to Johnson in 1964, so whatever racism existed, it wasn't extreme enough to go for Wallace. But Humphrey's potential in these states was limited at best: they weren't New Dealist enough to go for Kennedy in 1960. Both Humphrey and Wallace were far more hardline on New Dealism than JFK, and Humphrey was double so on social liberalism. Kentucky and Virginia had also been rocked by some of the worst of the 1968 riots-Nixon's highest quality cannon fodder against the Democrats. Therefore, the only former Confederate states that Nixon obtained in 1968 than he didn't in 1960 were the two Carolinas. There definitely was a deliberate appeal there to racial sentiment-you don't win in South Carolina politics otherwise. But they only counted for 20 electoral votes combined: without them, Nixon would still have 281, which was still enough to win.

And I'd argue that they were more likely to go for Nixon than not from the beginning: North Carolina was another "border state" that went for LBJ in 1964, so Wallace was a non-starter. No matter how progressive his economics were (or, in the case of middle class voters, because) or how satisfying his attitude with the hippies was, his image by 1968 was standing by the University of Alabama, and that put definite limits on how many states he could pick up. But it was definitely conservative enough (I've spent a fair amount of time in NC) so that by 1968 they'd had enough of the Great Society and riots and hippies. As for South Carolina, Nixon was very lucky that Strom Thurmond, while a racist hypocrite, was no political fool. He was interested in who could realistically win an election first and foremost after 1964. Nixon's private pledges to him on conservative Supreme Court justices helped, but I'm not sure he would have gone for Wallace anyway. There was a class conflict here: Thurmond was after the interests of wealthy Southerners, Wallace of poor ones. Thurmond would have been cognizant of that. I've occasionally heard it posited that Reagan would have won the nomination if Nixon did not do that, but I'm skeptical of that: Reagan was still very politically inexperienced compared to Nixon, 1968 America was still a far more hostile place to Goldwaterite economic ideas than 1980 America would be, and he also had problems elsewhere, such as his open escalation stance in Vietnam. Like Daley on the other side, Thurmond wasn't God-he couldn't just ignore the rest of the party. So, in total, no, Nixon did not need a "Southern Strategy" to win in 1968. It's worth remembering that at the height of his campaign in September, Nixon was going through the streets of post-riot Chicago, not South Carolina, which leads into my second point.

The "appeal to latently but not Wallace style racist middle class white people" strategy, on the other hand? No, Nixon would not have had a prayer in hell without that. He needed to separate the middle class from the working class, helped by the fact that 60s America was still an upwardly mobile place and many white working class people identified as to-be middle class. The two are often confused, not least because racism and Civil Rights backlash was far from an all-Southern phenomenon, nor was it all racism: law and order and crime were very valid concerns in 1968, and Democrats paid electorally for refusing to recognize that. Nixon was ultimately the founder of what Reagan would use with a few tweaks. The big appeal of Nixon that year was law and order and backlash against the Democrats. People thought that they were paying more attention to rioters than law abiding citizens like them. There were more states that Nixon won in 1968 that he didn't in 1960. They weren't in the South, but it was for the same reasons that people typically attribute to the "Southern" Strategy. Look at Missouri and Illinois, both of whom Kennedy won in 1960, but Nixon won in 1968. (I suppose one could make the argument that Missouri is a Kentucky-esque border state, but Illinois is definitely prime Union territory, complete with world city.) But I can't really see Nixon going against this anyway, they were his natural constituency. That's what he himself emotionally identified as. Nixon did see the long term potential of the South when few others did, long before the 1968 election. But what got him into the White House in 1968, for practical intents and purposes, wasn't really a "Southern" thing so much as a "non-bicoastal elite" thing. Again, why would Nixon-for all his faults and evils, nobody ever called him stupid for a reason-sacrifice pan-national appeal just to appeal to states that were probably going to go for Wallace anyway?

As for your last point, they fit in perfectly: what was the point of wasting electoral votes and not tapping into a massive source of them? Back in the 80s, the country was much whiter than it is now. It was only in the 90s where the GOP-and for that matter, the Democrats-stopped being an all-national creature. The South went back to Carter in 1976, and then went red for good in 1980 on an electoral level. On a legislative level, it wouldn't turn predominantly red until the Clinton era, and only under Bush II/Obama did the Dixiecrat finally become extinct. But race wasn't the defining issue of campaigns by the 80s, as opposed to 1964, even in the Deep South. It was there, but it was far from the deciding factor. Nationalism, social conservatism, economic conservatism, that counted too. Carter was a Southern New Democrat that was plenty social conservative, and of course, the GOP under Reagan was a perfect fit. The Nation and Company love to quote the 1981 Lee Atwater interview, but they rarely include the paragraphs after when he details that Reagan didn't need a Southern Strategy to win, either. I'm not arguing that Reagan didn't use race-I'm arguing that this wasn't the decisive factor in his success. That was disastrous in the 80s for the Dems-by refusing to come to terms with the fact that they lost the White House in landslides for perfectly valid reasons, not all of them race-related(the DNC commissioned a study in 1984 to find out why they lost so badly in three of the last four elections, but suppressed the results when it came back showing how out of touch they were. In many ways, it reminds me of the cognitive dissonance that the GOP Establishment shows right now), they were in the wilderness until Clinton.