Join Date: Aug 2009
Location: Portland, Oregon
From the Imperial Guard Codex, fifth edition. I'll take some quotes:
"Storm Troopers form the Imperial Guard's best fighting force. They are ruthless, proficient killers whose combat skills within the Imperial Guard are second to none."
This is important as it says that they're part of the Imperial Guard.
"...Storm Troopers are better armed and armored than regular Guardsmen."
This implies that there are other Guardsmen who are better armored than normal guardsmen--Storm Troopers, for example.
"Storm Troopers are constantly moving from one warzone to another and are amongst the most experienced units in the Imperial Guard."
It says explicitly that they're part of the Imperial Guard there.
This isn't to say that Storm Troopers can not be seconded to individuals or other Imperial bodies: the Inquisition, for example.
I think this disagreement stems from one misconception you're holding. Storm Troopers may have some duties that they share with military police--guarding important officials or positions behind the lines. But as you said, Storm Troopers do not do many of the duties expected of MP. Which is fair, since they're different.