Technically, there's not too much that's vital in the anterior part of the head. No major arteries or veins, and as long as your nasal cavity / pharynx remains clear, you can still breathe (the diaphragm is responsible for the physical breathing motion.) Of course, you're probably going to be breathing in tons of bacteria / blood, so unless you're treated quickly, time to say good night.

But yeah, that cow probably died from either sepsis or blood loss.