Is it safe to eat veggies that were chopped on bloody cutting board and then cooked?
Your girlfriend is right (lucky you). As long as food is going to be cooked thoroughly after coming into contact with the blood on the cutting board there is no risk as the heat of the oven will kill off any microbes that can cause foodborne illness.

The concern would be cross contamination of food that will not be cooked, for instance salads, breads, etc. This is why good professional kitchens are strict about washing boards after being used - in a busy kitchen it would be easy for mistakes to be made. In a home kitchen when there's good communication or just one person cooking you have much less chance of that happening. However, it would be all too easy to forget and crunch down on a piece of raw broccoli or for a small child to get exposed by raiding the uncooked vegetables, which is why it's good practice to wash boards after cutting meat even in a home kitchen.