Why shouldn't it bother me that there's no continuous carbon fiber running through/across the new joint?
Poorjohn
In answer to your question.............Because a properly executed scarf joint replicates the load transfer property's of the original structure
One way to think of this is to consider 2 hypothetical joints:
At one extreme would be a "butt" joint with resin (glue) in the gap, in this case there would be no fibers "across" the joint.
BTW: Does not matter if the resin or composite is stronger (actually stiffer) this creates a discontinuity / stress concentration aka not a good thing.
At the other extreme is a joint made by taking two barrel sections and tapering each from full thickness to none over the full length so one exactly fits inside of the other.
In this case there will be continouse fiber over the entire joint, although none transitions from one section to the other.
A scarf joint is similar to the second case expcept for the overlap distance.
BTW: Although I have some understanding of stress and transitions etc I am not a composites expert (at all) so feel free to correct/clarify the above. I am writing this partly to clarify my own understanding.