Why limit the discussion on design to teeth?

Jaw & Facial muscles:
Carnivores have evolved have wide mouths in relation to their heads and reduced facial musculature- it's an aid in killing prey. We've got a mouth cavity and jaw muscles made for chewing, like many herbivores.

Mammalian carnivores don't use saliva.
Human beings have carbohydrate-digesting enzymes in their saliva, which might suggest that we're adapted for eating all those animals made of carbohydrates, like the uh, er... um...

Digestive tract:
Carnivores have a short and smooth colon. Stomach volume of a carnivore is 60-70% of total digestive system capacity, with a stomach pH of 1. Their digestive tract is designed to extract nutrients quickly and expel waste quickly before putrefaction sets in. Humans have a small intestine about 10 to 11 times body length, a stomach pH about 4 or 5. Our digestive tract is adapted to hold food in for long periods of time to maximize the extraction of nutrients (from difficult to break down molecules like cellulose from plants).

Liver and kidney:
I don't pretend to understand the significant difference between bile secretion in carnivores vs. herbivores. I do know that carnivores use the enzyme uricase to break down the non-metabolizable uric acid found in flesh. Human beings lack this enzyme, and absorb and excrete uric acid with our kidneys.

While behaviorally primates may be omnivores, the physiological evidence ("design") suggests that they are herbivores, or more specifically, frugivores.