Being late with a great story is still being late.

A good explanation can make lateness feel settled while leaving the other person exactly where they were, waiting. The story is entertaining, and the entertainment does a lot of work in letting everybody skip the part where somebody stood outside for twenty minutes. The facts of the morning do not change with the telling.

Try dropping the story next time and saying only that you are sorry and you will leave earlier. Then leave earlier. If the reasons are always different but the outcome is always the same, the pattern is what needs fixing.