Geese can be good weeders, but in a vegetable garden it is very dubious.
In general, geese will always eat young, germinating grasses and shoots. So in a corn patch where the corn is 4' high, they'll walk down the rows and eat the weeds. However, they will eat leafy greens and brassicas of any age. So, they will go straight for your lettuce patch and go ape-shizzle before you realize what happened.
Also, my geese will work as a little team and will pull down corn stalks together and eat them (stover and all). So, you really have to keep an eye on them. The best use I have found for my geese is to leave them in the potato patch. They don't like the potato leaves at all. Corn can work. Pumpkins, zuccini, peas should all work.
But, they will find your spinach and that will be the end of it.