effectively, a president (or other high-ranking elected official) has to dress somewhat sloppily lest he offend his constituents.
I don't think Ronald Reagan ever dressed sloppily while president; not when there was the slightest chance of a camera around. Even when he wore his fake cowboy gear, it was never sloppily put together. Harry Truman wouldn't have been caught dead in a sloppy ensemble. Even Nixon rarely dressed sloppily while President; the infamous beach photo is one of the rare exceptions.
Maybe by "sloppily" you just meant "not as elegantly as was socially acceptable at the time." In that case, I agree. I do agree that Reagan and Nixon did not always dress as elegantly as they might have. But Manton and many others have frequently (and IMO correctly) said that Reagan, while president, was almost always well-turned-out.