I suspect that McAdoo's oil connection was more an excuse for denying him the nomination than the real reason. The Northeast was uncomfortable with his staunch Prohibitionism and his rural South- and West-oriented politics--including his unwillingness to condemn the Klan by name. (Actually, McAdoo did not share the Klan's anti-Semitism or anti-Catholicism, and he was no worse on African Americans' rights than many anti-Klan Southerners like Underwood. But he was in a real dilemma, because he was so dependent for support on delegations dominated or at least influenced by the Klan.) The Northeastern, urban, conservative, "wet" Democrats did not have the votes to nominate a candidate wholly to their liking but under the two-thirds rule they did have enough votes to block him.
My guess is that he should have at least mildly criticized the Klan--it would probably still support him because it really had nowhere else to go. (Except maybe for Senator Samuel Ralston of Indiana, but he ruled himself out because of poor health.
https://en.wikipedia.org/wiki/Samuel_M._Ralston)
Anyway, suppose through a combination of avoiding the oil scandal and deflecting the Klan issue, he does get the necessary two-thirds--or the convention, fed up with the deadlock, decides to abolish the two-thirds rule. One advantage he would have over Davis is that the railroad brotherhoods liked him (because of his record running the railroads for the government during the World War). He might even get some African American support--he had ordered that blacks and women employed by the railroads get equal pay with white men for equal work. (The all-white railroad brotherhoods didn't mind that--they figured that if the railroads had to pay blacks equal wages, they would be less likely to hire them. But regardless of that possible effect, and regardless of what McAdoo's motives may have been, the order was applauded by African American leaders.)
Would the position of the railroad brotherhoods have led to a collapse of La Follette's wider labor support and therefore perhaps of the La Follette candidacy? Jack Ross in his history of the Socialist Party doubts it: "Some historians have pushed the erroneous notion, based on the position of the Railroad Brotherhoods, that there would have been no Progressive candidacy in 1924 had William McAdoo emerged as the Democratic nominee."
https://books.google.com/books?id=fud1BwAAQBAJ&pg=RA1-PR21 But let's suppose that' notion is true, and assume the ideal situation for McAdoo--no Teapot Dome connection and no La Follette candidacy. Could McAdoo then have won a two-way race with Coolidge?
My answer is no. Let's assume that McAdoo wins every OTL Davis vote and every OTL La Follette vote. That last is very dubious. La Follette won many German-American votes because he had opposed the War. McAdoo by contrast had been part of the hated--by German-Americans--Wilson administration. Moreover, if La Follette doesn't run, the Socialists will presumably put up a candidate of their own, and he will presumably get some of the more "radical" La Follette votes. But let's nevertheless assume McAdoo can combine all the Davis and La Follette voters. He still loses. In 22 states with 296 electoral votes Coolidge got more votes than Davis and La Follette combined. In fact, if Coolidge were only to carry the states where he got 55 percent or more of the vote--against two opponents!--he would still win 18 states with 269 electoral votes--three more than needed to win.
https://en.wikipedia.org/wiki/1924_United_States_presidential_election
So to win McAdoo has to get every Davis vote (conceivable--but some conservatives may have supported Davis who would not support McAdoo), every La Follette vote (very doubtful) and even a substantial number of Coolidge votes. There is one peculiar consideration that could theoretically give him a few Coolidge votes. One of Coolidge's themes was "Coolidge or chaos." The idea was that if Coolidge didn't get a majority in the Electoral College, the race would go into the House where it might be deadlocked. If that happened, the vice-president chosen by the Senate would become president. According to horrified Republicans, that might be Davis's running mate Charles Bryan (portrayed as a half-witted radical) thanks to a Senate alliance of Democrats and "farm bloc" Republicans! Hence the GOP slogan (one of the most bizarre in US political history): "A vote for Davis is a vote for Bryan, a vote for La Follette is a vote for Bryan. A vote for Coolidge is a vote for Coolidge."
https://books.google.com/books?id=ltusveXdODUC&pg=PA212 This argument would of course not be available in a two-way race, but whether an argument so convoluted really added much to Coolidge's vote may be doubted. Anyway, I certainly don't think the absence of this argument would put Coolidge below 50 percent in any state where he got 55 percent of the vote or more--and those states, as I have already mentioned would by themselves be enough to elect Coolidge.
The basic problem McAdoo would face (apart from the divisions in the Democratic Party) is the same one faced by La Follette in OTL--they were counting on rural discontent, but 1924 was probably the best year for farmers during the 1920's. And Coolidge was seen as a man of probity, so it was hard to make the Harding scandals a convincing issue for voters.