There's been at least one conspicuously overweight president.
And for all we know there have been plenty of "unhygienic" presidents. How would anyone have found out if Lincoln never bathed or Hoover had bad breath?
Agreed but of course those were all before World War II. IIRC, the American obsessions with hygeine, grooming, and style in general came about during the war with the beginning of mass merchandising of deodorant.
Also, at one time, being obese was considered a "good" thing; it meant you could afford food, and plenty of it. IIRC, that also changed around WWII. Of course, now with carbohydrates being so cheap they're literally free (food pantries/soup kitchens serve up a lot of sugar and starch), weight tends to be in inverse proportion to wealth.
Although "image" is important, I do believe a "bad" image can be overcome with enough skill and the right message at the right time and place. If someone could do for economics during a depression what Einstein did for nuclear physics in the runup to a world war, he could probably roll around in feces and vomit for all anyone cared. Please note I'm talking about a damn near miraculous amount of skill, close to Strangerverse levels.