Just found your site, it's very informative so far.

I have a lake/pond approx. 9 acres. The depth of it at deepest is around 16 feet, I'd say. It gets a LOT of run off. This lake filled completely in 2 days. Hard as that is to believe, it's true. It is about 15 years old. Up until the last 2 years there has been virtually no top water plant life. We pump water from the lake to livestock.

The plant life that I have now is a dense rooted plant with elm shaped leaves on top of the water and a green algae.

I have been treating it in small doses over the summer with copper sulfate for the algae and weedtrine (?) for the leafy moss. These two things do kill the plant life but I would like to avoid having to do this at all.

The coverage was the entire shoreline anywhere from 10-60 feet out from the bank.

Since I'm pretty much ignorant to any kind of pond management, I need some good layman's advice. Would aerating help with the top water plant life?

When fishing a line down to the bottom, the fish that were caught were normally dead on the line. (lines checked after being out all night) That's what led me to think about aeration. I figured there was little or no oxygen at the bottom.

What am I going to see if I do install a system? Idealy, I'd like a lake that has relatively clear water, decent fish population and zero top water plant life. Is this a reasonable goal?

There aren't a lot of people who fish here. A dozen at best.

Just some basic questions about aeration I guess. Like I say, I'm new to it so if you need more information from me, just ask.

Solid questions, all of them.
Here's my thoughts.
First, the plants are never the problem, they are only the symptom. The problem is that you have sunlight, warm temps and food...all at the same time. Take away any one of the "Big Three" and plants don't exist.
That's why you don't see plants during winter. Too cold. That's also why you don't see plants in deep water...too dark, no sunlight.
Plants aren't a problem, when they exist in moderation.
Aerating won't do any good to prevent these plants. But, it will help to cleanse and oxygenate deeper water. The catfish you caught which died...they went below the thermocline to feed, and couldn't come back up because of your trotline. They suffocated. Aeration will definitely help there.
Your pond is an excellent candidate for winter drawdown. Since it has such a large watershed, I would drop the water level this winter. Drop it five or six feet and keep it down during the coldest two weeks of winter. Winter's frigid temps and drying effects can kill much of that peripheral vegetation. Then, let the pond refill next spring, and you stand a fair chance of losing the plants currently causing the biggest problems.
I don't know if you can buy grass carp in Missouri, but they may be a viable option, too. The plant you describe is one of their favorites.

About the grasscarp, how do I determine how many and how big? There are some pretty large flathead cats in here, got one that was 32 lbs recently. I imagine I'd need relatively large carp to possibly avoid becoming food for flatheads.

I don't know the laws in Missouri, so first check with your DNR. Then, once you know if you can use them, there will be licensed dealers. Contact them and tell them your situation. Stock no more than 4 per acre in the beginning. Since you have big flatheads, you need to stock the biggest grass carp available. Several hatcheries south of you in Arkansas have them. Buy them later winter, or early spring. You may be able to buy fish several pounds apiece.