It's certainly possible to get a cutaneous infection that doesn't go systemic or end up in the respiratory system. For viral and bacterial infections.
For instance, you can get a cutaneous anthrax infection (a bacteria) and it's no huge deal if treated. You get nasty skin lesions and black ulcerations. With antibiotics your chance of dying is low.
A respiratory infection of anthrax on the other hand can frequently be fatal even it treatment is attempted. It progresses so fast that you can be dead within 48 hours of symptom onset.
As mentioned before, cutaneous smallpox infection has about a 5% mortality rate, and usually had fairly mild symptoms. Airborne infection by smalpox was fatal about 30% of the time, and if it didn't kill you it frequently left you seriously disfigured from scaring from thousands of lesions on the skin.
A lot of it has to do with how much access the infecting organism has to the major systems of the body and therefore how rapidly it can propagate. The first few layers of the skin are not heavily vascularized so the infection has a limited access to the bloodstream and spreading to the rest of the body and becoming systemic, it therefore spreads slowly. This allow the immune system some time to start making antibodies and mounting a defense. The respiratory tract on the other hand has extensive access to the bloodstream. Therefore an infectious agent can enter the lungs, quickly get access to the bloodstream, and can start reproducing throughout the body before the immune system has much time to start to react.