So, so far you don't have concerns about significant leaks through the dam and you don't plan to rework the dam for any reason? Trees/vegetation on the dam can be a problem down the road but it appears that is not the main concern now.

IF you are OK with the current depths and level of muck then it sounds like you are not so worried about rehabbing the dam or the depth or muck in the pond but just want to start over to get a better fish population?

Aeration can slowly help reduce muck over time and is beneficial in many other ways. Consider doing research on that.

IT sounds like your pond has watershed sources of water but if it is a gravel bottom then it may be a groundwater pond too with water flowing in and out through the gravel at the bottom. That would mean that you wouldn't worry about 'leaks' so much since it already leaks all the time but refills from watershed and rain events.

I guess you then focus on whether you want to remove large amounts of fish yourself to help restore balance or if you would like to use a chemical to kill many/most of the fish to help you start over?