Another civilization at the edge of our observable universe could be looking in our direction "right now" and make the same observation because the image that they would get is equally ancient.

That's correct! for them, our location would be a point on their CMB!

The CMB thus shows us how the universe was like ~14bn years ago since that it the estimated distance to its source.

Actually the distance of the source will be a little more due to the expansion of space in our observable universe during the 14 billion years of its existence. By taking it into account the rate of expansion (and considering it constant), it can be calculated that the radius of the observable universe is about 46 billion light years.

But if the entire universe is far greater than the observable universe, doesn't that mean that the universe could be much older than 14bn years?

No, all sections of the entire Universe were born at the same time. Within the scope of eternal inflation, if two different points change state at a different time, that means that the space between these points would have inflated extremely during that time differential. These 2 points would become two different universes, completely disconnected, with maybe different values for the fundamental constants (yet, still the same laws of physics)... That is called the multiverse of level 2 - I'll probably write a series about this one day ;-). So to conclude, the entire universe in which we live in is 14 b years old.

In other words, how do we know that the CMB shows the state of the universe 'soon' after the big bang.

The CMB shows the state of the universe 14 billion light years away from us (or 46 billion light years if we consider expansion), not beyond... hence my doubt about defining empirical theories on such a limited sample... It actually shows the state of that section (a surface) of the Universe 380000 years after the Big Bang...