Why Pass Ancient Policies? - One big reason is because it's very unlikely you'd be able to get the World Wonder they unlock upon completion, but generally they are designed to start a game off, and the following era policies help you in more effective ways.
Progress gives nice per city yields, for example, and encourages settler spam, but Piety does the same thing and also helps unlock a Religious National Wonder. Authority's aggression is also incentivized with Statecraft and Piety, but instead of just being stronger in war, you gain more religious or diplomatic power. Progress per citizen births are nice, but Aesthetics makes you a Tourism powerhouse, which will help you win the game.
Basically, they're the elementary school policies, you're in high school once you're done and should take high school policies.
Piety - The main advantage with Piety is that you can annex or settle an unlimited number of cities and be able to purchase religious buildings in them. Having religious supremacy, especially if you conquer rival Holy Cities, is a big boost in diplomatic standing too. If you take this build up your military, conquer and settle your way to max pressure.
Statecraft - Statecraft benefits from having a large number of cities too, but you don't have to annex a city for them to build Chanceries / Wire Service. It's about diplomacy, and city-states are all over the world, so you benefit from having an overseas city or two to grab up as many alliances as possible and maintain them with trade routes that bring in influence per turn.
Aesthetics - Aesthetics plays into the idea of having the fewest cities possible. Policy costs being as low as possible with all three Guilds in your cities is the fastest way to gain policies in the game. You also get Tourism per Wonder, which Tradition is excellent at building throughout the game because of an early Engineer and faith purchasing Engineers.
