What exactly does cooking meat do as far as sanitation goes?
Very simple: it kills bacteria. 

The recommended cooking temperatures are empirically derived. They lead to a log 7 reduction in the amount of bacteria present. That is when you have a piece of meat, 1 in 10 000 000 of the bacteria on it remain alive after cooking. The recommended temperature for storage and cooking is chosen in such a way that, for food which has been handled safely before cooking, this reduction is enough to produce food which will be safe for the next 3-5 days in the fridge. 

There is no path from "unsafe food" to "safe food". There is no telling how many bacteria are on your food if it was not handled properly. Bacteria grow exponentially, so small differences can result in not just a few more bacteria, but in many times as more bacteria. Also, there is always the problem of toxins created while there was a lively bacterial colony on the meat, even if you cook the bacteria to death afterwards. 

In short, food is only safe when it is handled safely before cooking, cooked safely, and handled safely after cooking. A slip at any step cannot be remedied.