I would keep separate "life of mathematicians" from "history of math".
While the first certainly contains much narrative material, which would make the subject more "attractive", its pedagogical content is, to me, doubtful.
As for the second, on the other hand, I am inclined to say that everybody would get something from presenting math as a lively subject; and lively means also that it has a history, containing evolution, wrong turns, errors, repetitions, trends and all the like.
Unfortuntaley, at least for what concerns my country, Italy, very little emphasis is usually, at high school level, put on such a presentation of math. The reason being that it would take time away from "real math". I think this is an error in perspective, since it mantains a status quo in which a large majority of students exit high school convinced that math ended somewhat in the 19th century with nothing new going on after that. As a research mathematician one of the most common questions I am asked is: "is there really something new to say about math"? (Guess this is common)
You mention Koningsberg problem. I devoted some time to think abouth Euler charachteristic in the context of polyhedra. If you browse the book "Proofs and Refutation" by Lakatos you may well find an enormous amount of informations, mathematical and historical, that could be presented in a class to show how that subject, starting from an apparantely simple remark open a whole new field of research for years.
I referred to the fact that I'd rather keep separate this from "life of mathematicians" since I fear that encouraging history of math in high school could result in simply adding to math some spicy story on people, anedoctes, dubious interpretations (like Cantor being driven crazy by infinity) which are ok for entertaining a general audience but not for class time (imho, of course).