Why does bed bugs not carry disease?
It's not entirely true.
For example, a 2014 study demonstrates bed bugs can transmit Chagas disease. From Penn Medicine News:


  The bed bug may be just as dangerous as its sinister cousin, the triatomine, or “kissing” bug. A new study from Penn Medicine researchers in the Center for Clinical Epidemiology and Biostatistics demonstrated that bed bugs, like the triatomines, can transmit Trypanosoma cruzi, the parasite that causes Chagas disease, one of the most prevalent and deadly diseases in the Americas. 


(I'll update with more details when I get off the road)