A shallow pond is going to have a higher percentage of water that supports weed growth. Not optimal, but you can work with that.
A shallow pond is also going to be hotter in the summer. Hotter water is less capable of holding dissolved oxygen. This puts your fish at risk of a fish kill. Once again, not optimal, but you can work with that.
Do you have AC electricity available at/near the pond? Reliable aeration would probably be a big plus for your pond.
If you stocked fatheads two years ago, it should be packed with them. Are you seeing minnows in the pond?
I do not have AC Power available. I did see minnows this spring / early summer and did have a what I believe was a spawning session where I could see them churning under some leaves but I haven't seen any latey. I put in three pallets to help the spawn but they have sank into the shallow water.
I do have a large number of crawfish and turtles. I'm not sure how hard these predators would be on the fatheads.
Interestingly, there is not a single weed in the pond. No surface scum, nothing?