You seem to be missing my argument. I recognise that there are cultural influences in wonders, which is why they are so. What if, as you complain in your OP, the only thing stopping me from building a wonder is the turn order? A perfect case in point (for a real example) are the Pyramids in Egypt versus Chichen Itza in Mexico. They were built in vastly different era and by two separated civilizations. They are, essentially, a similar structure, yet the greatness of one does not diminish the other. Translating this into gameplay: Why should someone building a wonder simply because they have the 'first player' token in front of them stop me from building the same wonder in the same turn?
Again, by categorising them as 'wonders' as opposed to 'the pyramids' or 'stonehenge' you are undermining your own argument. I could respond, along the same lines of logic, that 'you can always build a different building' or 'you can always train a different unit'. I doubt there will ever be a game where you won't be able to build a building or a unit. So again: Why are wonders accepted as 'one-of-a-kind' buildings but others not considered 'four-of-a-kind'? It's an illogical double-standard.
Equally, you say that not being able to get barracks or aircraft is a game-breaker, but surely this depends entirely on the dynamic of the game. In some games, not being able to get Uranium must be a 'game-breaker', or being unable to build the Sydney Opera House. If someone relies on building any specific building in their strategy, then being denied it will evidently ruin their strategy. The simple solution is to adapt your strategy to the dynamic. If it is clear that you are going to be beaten out when it comes to getting the barracks and aircraft, then research faster and hold the world ransom with nukes. Or ruin their games by culture-bombing and utterly screwing their plans.
The game deals with finite resources, it is how it was designed such that players must be sure to obtain these finite resources before they are depleted. Complaining about two specific resources whilst ignoring the rest is an incredibly odd dichotomy.