the conclusion you came to assumes that the universe was static.

The universe is actively expanding. Think of it as on a treadmill, or trapped in a current.

The light is traveling towards us, but the space is also expanding as it travels.

Here is a very simple set of numbers to help you see this.

You walk at 2m/s. You start 10 meters away from me. You are standing on a treadmill that moves at 1 m/s. You start walking. You walk for 1 second, you walked "2 meters" and normally would be at 8 meters. But the treadmill slid you back 1m, so you're at 9.

repeat, another second and you're at 8, then 7... it takes you 10 full seconds to reach me, you walk 20 meters total because the pesky treadmill undoes 1m every second.

When I observe you, I measure you to be 20 meters away, even though your original position was 10. This is because the light has spread out as if it traveled for 20m, redshifted over 20m, and such.

So the galaxy was far closer than 13 billion ly when the light first left. And indeed the amount the expansion matters changes (i.e. the treadmill isn't constant).