I don't think Mellon was ever interested in the presidency. By 1928, he was rather elderly, especially for that era when life expectancy was much lower than it is today. Mellon did a good job as the long-time Treasury secretary, spanning the terms of Harding, Coolidge, and (partly) Hoover. He can be considered as the father of what later became "supply side economics."
As it turned out, BTW, Coolidge died of natural causes three months before Hoover's term was completed, and very concerned about the future of the country under President-elect Franklin D. Roosevelt.
Having someone not interested in the Presidency is precisely the person you want in service. Considering that Mellon continued serving at Treasury under Hoover, and concluding as Ambassador to the Court of St. James (England) showed that he was able to serve (and he lived until 1937, so could’ve served 2 terms, had 1929 been a Panic instead of a Depression).