Regarding a similarity between Pontryagin classes and Chern classes, see this very interesting mathoverflow answer by Johannes Ebert: mathoverflow.net/questions/16632/… . However, that answer does not address any possibility of describing Pontryagin classes as Chern classes for some complex oriented theory.

@David: You are most welcome! Please let me know if you find any issues with my answer.

@PDC: It was my pleasure.

@Ronnie: Thank you very much for your comment. I had actually heard that before about the Hurewicz theorem in Spanier's book, but had forgotten about it. I must admit I have never checked that section of Spanier's book --- along with many other sections :). Also, I was not aware of the original references for the right properness of the Strom model structure. Thanks!