I've thought that too, and I think the answer would be it depends on how the executive branch would be elected (with one representative per district or a group of people nationally elected). If the people would be elected nationally, it would probably be like it is now (where if things are better they get re-elected) whereas if it were local representation it would be much like Congress. Despite Congress polling at something like 9% popularity, Congressmen and women laugh because those polls don't matter. Which matters, is how popular congresspeople are within their districts. My professor on Congressional and Presidential rhetoric said it best "Congress is nothing but a bunch of Washington-insiders and criminals. They should all be arrested! Except for my congressman, he's one of the good ones." This is the reason why despite such low popularity, incumbency is so high. So, if the executives were locally elected, I see it being much like this. I may be wrong (of course) but this is my take on it.
EDIT: And that's another thing that is unique to the President in America, they are the only person elected nationally, by every state. This could also play into the expectations of the presidency where people expect the President to make things good in general (economy, gas prices, unemployment), whereas representatives just have to "bring home the bacon" which is considerably easier a task.