Where do you get the idea that the reason it is not recommended on athletic uniforms is so it will not get soiled? It is certainly going to get soiled on police and military uniforms.
From the Flag Code as posted at ushistory.org
"j. No part of the flag should ever be used as a costume or athletic uniform. However, a flag patch may be affixed to the uniform of military personnel, firemen, policemen, and members of patriotic organizations."
in combination with
"e.The flag should never be fastened, displayed, used, or stored in such a manner as to permit it to be easily torn, soiled, or damaged in any way."
perhaps the framers of the code were able to figure out the exception for the military, firemen, police over-rides the rule to not allow it to get soiled. So you can argue all you want about the soiled thing, doesn't really matter. The bottom line remains, we as officials do not fit the catefories allowed, by code, to wear the flag on our uniforms.