Let's take as our POD that Theodore Roosevelt dies during his Brazilian expedition of 1913. The Progressive Party disintegrates even more quickly and completely than in OTL, and in 1916 the Republicans do not have to pay as much attention to TR's followers as they did in OTL. (Hughes, despite his weaknesses, was probably the best choice for the GOP that year in OTL because as a Supreme Court justice he did not take part in the TR-Taft fight of 1912.) They can go with their real preferences and choose someone more conservative. After a deadlocked convention, a "dark horse" candidate is chosen: Senator Warren G. Harding of Ohio, who has been in the Senate only since 1914. (There have been enough "dark horses" nominated in US history to make this plausible, and Harding would have support from Old Guard Senators despite--or even because of--his comparative lack of national political experience; they might think it would make him easier to manipulate. And of course, he is from a key state, Ohio.)
Had TR been alive, Harding would be very unlikely to be nominated; in 1912, Harding had been very much a Taft loyalist. Placing Taft's name in nomination at the Republican National Convention that year, he called the President "the greatest Progressive of the age." When TR decided to run as a third-party candidate, Harding's *Marion Star* called him "utterly without conscience and truth and the greatest faker of all time" and compared him to Benedict Arnold. But in this ATL with TR long dead, all this was forgottten as it would be in OTL in 1920.
And not only does TR's absence help Harding get the nomination in 1916; it also helps him win the election. TR's bellicose speeches were the leading reason why the Hughes campaign in OTL could not sufficiently capitalize on German-American (and to some extent Irish-American) resentment of Wilson's allegedly un-neutral conduct. With TR gone, Harding is free to exploit this resentment. In German-American areas, he stresses his determination to resist British as well as German violations of American neutral rights; in Anglo-Saxon areas of the Northeast he argues that Wilson's weakness encouraged the German submarine campaign. Yet he is careful to emphasize that he does not favor the US entering the war. His slogan is "Not Nostrums But Neutrality."
The result is that though Harding loses Progressive Western states like California (as did Hughes) he makes up for it by carrying his own state of Ohio (and perhaps Missouri, another state with a large number of German-Americans).
http://psephos.adam-carr.net/countries/u/usa/pres/1916.txt After the election, he announces his Cabinet choices, including Secretary of State Elihu Root...
What next? Presumably he will favor war after Germany's decision for unlimited submarine warfare, as he did as a senator in OTL. After Germany's defeat, I think he recognizes that going to Paris to negotiate the peace treaty would be something way above his head, and wisely sends Root there instead. The result might be a weaker League of Nations--one without Article X--but at least one in which the US will be willing to participate. The peace will still be unpopular, though--just about any conceivable peace treaty will offend large numbers of Americans [1]--and the end of the war will also bring economic problems. (There is going to be inflation--which people will blame on the "profiteers" [2]--followed by a necessary but unpopular deflation.) For these reasons, I don't see Harding getting re-elected in 1920. But despite the likely scandals in his adminstration, neither do I see him losing so overwhelmingly as Cox did in OTL. (For one thing, the Republicans are less likely to be blamed for getting the US into the war than the Democrats were in OTL; Harding can plausibly blame Wilson for having left the US in such a state that war was inevitable.)
(Root, whose advice Harding would presumably take, was not against the idea of some sort of League--but he and many other Republicans thought Wilson's Article X was impractical as it represented a US commitment to defend the territorial status quo everywhere. What Root wanted instead was a treaty with France guaranteeing it against future German aggression; and this, being much less open-ended than Article X, might have a better chance of passing the Senate.)
[1] "It's too harsh on Germany." "No, it doesn't give Poland enough." "Why are so many other nations guaranteed their freedom but not Ireland?" "It doesn't give Italy enough." "No, it gives Italy too much at the expemse of the Austrians (or South Slavs)." Etc...
[2] Because of the association of Republicans with big business, that might hurt Harding even more than it did Wilson. Profiteer-bashing may well have been irrational, as Keynes thought: "In the latter stages of the war all the belligerent governments practiced, from necessity or incompetence, what a Bolshevist might have done from design. Even now, when the war is over, most of them continue out of weakness the same malpractices. But further, the governments of Europe, being many of them at this moment reckless in their methods as well as weak, seek to direct on to a class known as 'profiteers' the popular indignation against the more obvious consequences of their vicious methods. These 'profiteers' are, broadly speaking, the entrepreneur class of capitalists, that is to say, the active and constructive element in the whole capitalist society, who in a period of rapidly rising prices cannot but get rich quick whether they wish it or desire it or not. If prices are continually rising, every trader who has purchased for stock or owns property and plant inevitably makes profits. By directing hatred against this class, therefore, the European governments are carrying a step further the fatal process which the subtle mind of Lenin had consciously conceived. The profiteers are a consequence and not a cause of rising prices...."
https://www.pbs.org/wgbh/commandingheights/shared/minitext/ess_inflation.html But rational or not, such hatred was widespread. Frederick Lewis Allen imagined a middle-class couple in 1919: "Mr. Smith tells his wife that 'these profiteers are about as bad as the I. W. W.'s.' He could make no stronger statement."
http://xroads.virginia.edu/~hyper/allen/ch1.html