You're wrong about the Presidency. The Constitution explicitly states that he or she must be a natural-born citizen of the United States; naturalization won't

You're wrong about the Presidency. The Constitution explicitly states that he or she must be a natural-born citizen of the United States; naturalization won't do. State election boards generally take the word of the political parties, but as has been shown in recent years eligibility can be challenged in court -- most often by racist shitheels upset about the skin tone of the candidate in question. A qualifying birth certificate is usually the path taken here. So yeah, you don't have to show it unless challenged, but if you can't show it you likely WILL be challenged, and that could be a problem.