> If it expanded faster, the light would never catch up. I doesn't seem like we should be able to see anything at all.

The mistake here is that you dismiss the fact that speed of the expansion is related to distance. What happens is: "space expands". Imagine that 1m of space at some point becomes 2m. This also means that 100m become 200m in the same timespan. Notice that this means that object which was 1m away is now 2m away (so moved away by 1m) but object which was 100m away is now 200m away (so moved away by 100m).

So while expansion makes everything further away from everything else, the distance change is greater the further the object is. So objects which are closer are getting away slower, and objects which are further are getting away faster (and even faster than the speed of light!).

It's true that light from things very far away won't ever reach us, because space expands faster than the light can travel, but there are lots of objects closer than that, and light from those objects will eventually reach us.