A follow up. I appreciate all the advice, and I did look into some kind of excavation or dredging. Decided against it as the contractor warned me that it could create more leakage in the pond bottom. Leakage comes out in the neighbors horse pasture and makes a muddy mess. And it was expensive.
So I tried fish, I stocked catfish, bluegill, mosquito fish, crawfish, and snails back in about March. Now the algae is almost completely gone, I went from 95% algae covered mess to less than 5%. Really a miraculous change, it has never been so algae free.
The only other change in the pond has been due to our drought, probably the driest year since the ice age. Our spring stopped flowing for about a month and the pond dropped a bit, but then flow started up again albeit slower than normal. According to the neighbors its the first time it has dried up in anyone's memories. The spring is regulated by the state of Utah water rights people, they rate it at 0.5 cfs. Farmers downstream use it for irrigation, but not this year.
Cat tails are doing well, probably too well, but they give the fish a place to hide and we don't mind them.
The pond is about 75 ft in diameter and 8 ft at the deepest point. Most years the spring flow keeps the temperature close to 55, but this year due to low flow it has warmed to about 75.