Not the slightest chance of Pershing. First of all, in 1936, a 76 year old candidate for president was extremely unlikely; no president had reached the age of 70 even at the
end of his presidential service until Eisenhower in 1960. Moreover, Pershing's political weaknesses were evident even in 1920:
"But in mid-April, a nationwide poll by the
Literary Digest, showed Pershing running a distant ninth in a field of 14 potential Republican candidates. Perhaps even more discouraging, the top spot in the poll was held by another Army hero: Major General Leonard Wood. A medical doctor by training, Wood had made his name in Cuba during the
Spanish-American War, where
Teddy Roosevelt, the former Republican president, had served under him. Though Pershing’s accomplishments were more recent, and he might have seemed like the fresher candidate, he and Wood were about the same age; in fact, Wood was born a month earlier.
"Part of the reason for Pershing’s poor showing in the polls, some commentators explained, was that as a firm, by-the-book general often described as “unsmiling,” he was respected but far from loved by what might have been his natural constituency: his former troops. They and their families would be making up a substantial chunk of the electorate that November.
"A writer for
Munsey’s Magazine, a widely read periodical of the day, tried to put it diplomatically. “He has much of the glamour that surrounds a victorious general, he unquestionably possesses high ability, and physically he is a hard muscled veteran of fifty-eight,” the writer noted, starting on the positive side. However, he added, “if what the returning soldiers… say is true, General Pershing is not to the American Expeditionary Forces exactly what Grant was to the Union Army. The admiration is there, but not the measure of affection which the Northern soldiers gave to Grant.”"
https://www.history.com/news/john-j-pershing-presidential-campaign-world-war-i
And in any event, in 1936 the last thing voters wanted to be reminded of was the Great War.
As for Ruth, first of all he was a Catholic which only eight years after Al Smith's landslide defeat would have been disqualification rnough, and second athletes (like actors) who want to go into politics do not
start with the presidency if they want to be taken seriously. (Sure, businessmen with no experience in public office have been nominated but that is different--many voters see the government as analogous to a business, and anyway both Willkie and Trump had been outspoken on political issues for years or decades before their presidential candidacies.)