When baking meatballs for a soup, do I have to bake until the meat is fully cooked?
It doesn't matter in which modality (banking, simmering) you bring the meatballs up to temperature, only that you do.

Normally, if you are cooking them in the oven first, it is for browning generate additional flavor.  Otherwise, you could simply simmer them from the beginning, which is perfectly viable.