I doubt if changing the Republican candidate will make much difference. Harding was ideal. After the tumultuous events of the last few years, the blander the better.
OTOH, if the Democratic nominee is McAdoo or someone else closely identified with Wilson, then Democratic turnout might drop, giving the GOP a higher percentage of the popular vote, but not necessarily an increase in their absolute numbers.
I doubt that McAdoo or any other plausible Democratic candidate (except Al Smith if you consider him plausible) could have done worse than Cox. As I wrote here a while back:
"Granted that the Democrats' cause in 1920 was hopeless anyway, Cox was an astonishingly bad candidate. In his last campaign address he denounced African Americans and German Americans for supporting the GOP and announced that "Every traitor in America will vote tomorrow for Warren G. Harding." "People walked out of the hall."
https://books.google.com/books?id=Uia4A04q8dMC&pg=PT378
"(Cox was incidentally just about the worst candidate for getting any German-American votes. As governor of Ohio, he insisted that teaching in German be banned in private and parochial schools as well as public ones, over the opposition of the Catholic Church. Then he chose FDR as his running mate--and German Americans rightly or wrongly assumed that this Roosevelt shared the views of the other one toward Germany. Small wonder Cox got 16.17 percent of the vote in WI--not all that far ahead of Debs!--18.19 percent in ND (which Wilson had actually won in 1916) etc.
https://en.wikipedia.org/wiki/1920_United_States_presidential_election)" (Of course no Democratic candidate was going to get very many African American votes, but at least McAdoo would have his record as wartime administrator of the railroads where he ordered the railroads to pay African Americans and women equal pay with white men for equal work. This was actually done at the request of the railroad brotherhoods who no doubt reasoned that if the railroads had to pay blacks equal wages, they wouldn't hire them. Nevertheless, McAdoo's order was applauded by African American leaders. McAdoo would also probably have had stronger labor support than Cox.)
Also, Cox had the reputation of a relative conservative (despite a mildly progressive record as governor of Ohio) and a "wet" on Prohibition which hurt him with Bryanites in the West (McAdoo might have done better there, though probably without winning any electoral votes in that region) without really helping him in the Northeast. Some ethnic voters were glad that the Democrats had nominated Cox rather than McAdoo whom they associated with Wilson--but as soon as Cox came out for the Versailles Treaty, they concluded he was just as bad. As a study of the politics of Providence, RI's "Little Italy" notes:
"In view of Wilson's stubborn rejection of their ancestral county's claims, Italian Americans welcomed the defeat of the president's son-in-law and apparent political heir, William McAdoo, in the contest for the Democratic nomination for the White House and were initially satisfied with the selection of James M. Cox as the Democratic presidential candidate. Such an early approval suddenly turned into disillusionment as soon as Cox declared that he adhered to Wilson's vision of U.S. foreign policy. He referred primarily to the acceptance of American participation in the League of Nations and did not elaborate on his stand on the issue of Fiume. But, in the eyes of many Italian Americans, his words were sufficient to make the 1920 presidential race into a sort of referendum on the lot of the Croatian port or, at least, to perceive that year's elections as an opportunity to retaliate against the incumbent president for his disregard of the interests and wishes of the Italian government. As
L'Eco del Rhode Island put it, "the major difference in the two platforms is to either accept or destroy Wilson's policy. It goes without saying that, deep in our hearts, all we Italians are ready to fight for the destruction of his false concept of democracy." Similarly,
Il Corriere del Rhode Island remarked that "Cox's statement is enough to lead the Italians to resort to all their strength in order to oppose Cox and all the other candidates who reveal even the slightest support for Wilson's programs. The Italians must not forget and will never forget that, if Italy—after her heroic and titanic efforts—is still struggling in the tentacles of foul Molochs who are trying to stifle her with their voracious speculations...""
https://books.google.com/books?id=vtP0KA_L9WAC&pg=PA52
In the very unlikely event that the Democrats nominate Al Smith, they will undoubtedly do worse than Cox did in the Electoral College (though not necessarily in the popular vote) because Smith is almost certain to lose some southern electoral votes Cox got (KY definitely; NC and some other states very probably), while he is unlikely to carry any northern states, even the most Catholic ones like RI or MA, though he will certainly improve on Cox's showing there).