The Designated Hero and Protagonist-Centered Morality tropes that are brought up on this page are actually justified in-universe: early in the film, Alexander describes to Dave why what they are doing is not wrong. It all comes down to intentions: if you aren't hurting anybody, and everyone stands to benefit, then the law can be ignored (the specific argument used is, if you were driving down a road with your deathly sick mother and came to a red light where no traffic was present and you knew it was totally safe, would you run the red light?) The comparison becomes valid later on, when Alexander's and Mitchell's extensive illegal actions come to light. Essentially, while Dave's actions are no less illegal, they are more moral and righteous. Whether or not this is a bad thing in and of itself is up for debate, but when held up against Alexander's actions, it's the superior choice.
When one considers that Mitchell's campaign probably depicted him more like well-meaning, compassionate Dave than like his scheming, self-serving true personality, one could also argue that Dave is giving America exactly the sort of President it thought it was voting for.
Also, if they did denounce the substitution, it would be even worse. Essentially, they would deal the death knell to the people's trust in the government, creating a huge headache for everyone. Dave simply minimized the damage by hiding the conspiracy in a way that the only repercussion was the destruction of Alexander's career.
At the end of the movie, the recently-widowed First Lady starts up a romantic relationship with Dave, a man who looks a lot like her deceased husband, and can't account for his whereabouts over the past few months. You think that might make people a teeny bit suspicious?
Eh, people will assume she's also an impersonator.
They might also assume it's something to do with her grief over the death of her 'beloved' husband; a bit unusual, perhaps, but not outside the realm of possibility.