I have started to see more and more of incoming goods risks being considered. Failure to do this in some form (whether you chose to have it as part of your HACCP plan) can lead to disaster. One time we were launching a new product which contained raw, ready to eat ingredients. Due to how busy the factory was, I was told to leave that project to my boss. Unfortunately, the boss didn't consider the risk of ingredients so we had herbs which were not meant to be ready to eat going into a ready to eat product! I discovered this just in time; however, this meant the costing was wrong on the product.
The other thing to think about is the supplier isn't necessarily checking the hazards for you. If you use creme fraiche as part of a cooked sauce, you won't be as worried about the micro as if you were using it in a salad dressing. The supplier doesn't necessarily know what you are using their products for.
I wouldn't assess all ingredients separately, I'd group them, e.g. washed vegetables, dirty vegetables, raw chicken. You can then assess easily what the hazards are, whether there are any special cases (e.g. celery allergen or higher risk of bones in chicken thighs) and use this as a basis for the supplier specification going forward. It will also help you to decide whether you need to positive release any ingredients coming into the factory based on results. It would then be really useful for auditors to know the expectations from the supplier in this hazard analysis, preferably before you launch but at least during routine audits.
I would start this with new product launches and gradually backdate if you can.