Coordinates don't mean anything. There is not necessarily a coordinate that is "the time coordinate." If you have a coordinate whose gradient happens to be in a timelike direction, you could say it's a timelike coordinate, but the finiteness of a change in such a coordinate doesn't necessarily have any physical significance.
If we integrate over the time coordinate t and find that it takes a finite amount of time for light to travel to r=∞
It doesn't have the interpretation of time to travel.
and then when we integrate over an affine parameter τ it takes inifinite amount of time to travel to r=∞
Nor does this have the interpretation of time to travel.
My (GR) intuition tells me that there is a "coordinate" singularity at r=∞(for coordinate τ).
This affine parameter is not a coordinate.
In the situation you describe, the interpretation is probably that $t$ just doesn't blow up very fast as you approach null infinity. That's not something that has a physical interpretation, it's just a fact about your coordinate system.