Woah!...The correct way of looking at these kinds of questions is from the perspective that there is something in the theory or physics that YOU do not understand and that the experts do. Any apparent confusion emerges from the novice's not understanding the physics, not from the astronomer or physicists having made some idiotic error.
In this particular instance, you have assumed that the universe is a Newtonian system, like the fireworks display you see on the 4th of July. The expansion age, even of such a fireworks display, can be found by measuring the speed of any part, and its distance from the center of the explosion. It doesn't matter which part you choose. But the universe is not a Newtonian system even though our common sense insists on framing questions and insights as though it were. But at least from our point of view, the correction to the relativistic model is only a slight one so long as we do not look too far out in space. Because the expansion speed is linear with distance, it doesn't matter if you look at a distant quasar or a moderately nearby galaxy. It doesn't even matter if you 'look' at galaxies you cannot even see yet because they are so far away. The linearity of the expansion speed with distance means that ALL of these distance-speed combinations will scale exactly the same way to give you the same value for the expansion age of the universe.