You've mentioned that the water flows up and out the ground even going below the surface on your own property to come out in another place on an adjoining property. Because of this, I am inclined to think its pretty rocky around there, and especially under the ground a short distance. The preponderance of springs in your area tells me the same.

So I wonder why there is a pool there at all. If I were to guess, I would say that the reason is that in the past this was a very permeable spot where to water flowed, at least part of it, into the ground. This flow, I propose, had the effect of washing away the looser portions of the soil and caused a sink-hole of sorts ... right there. So what does that mean? Well for one thing it may mean that the muck is actually what is helping to keep the pool at full pool.

At this point it is difficult to say whether you could remove all of the muck and still have a pool that holds water. If so there may be ways to substitute the muck with another sealant. River silt, the kind that forms visible particles that or of the order of size 80 sand might be ideal but it would depend on the size of the fractures that carry water away. You might use soil on your property but for it to work it must form a bridge. On the other hand, you may find that you can take all the muck out and that these fractures are already stably bridged and the pool holds sufficient water with the influx of the spring.

So how to get the muck out. IMO, the benefit you have is that there is a good flow of water. Because of this you can dredge the muck and if you have time to spend with it you could do this yourself. I would use something like 4" Poly tubing of sufficient length to access all parts of the pond and enough extra to get the outlet to a lower elevation. The greater the elevation difference the better this will work. Charge the tube with water by completely submerging it (start one end into the water and continue until complete. You need some way to prevent air coming in when you lift the outlet of the charged siphon from water (like a full opening ball valve ,.. the valve must be open while charging the tube). If you take this to a lower elevation and open the valve, the water will flow on its own. Because the muck is soft, it will be drawn into the siphon and discharged at a lower elevation. Think about where it is going and how it will affect you and neighbors. It may be that a smaller tube would be better (for example 3") in that it may be possible to set it up so that it is moving only as much or less than spring produces (which would mean that your dredge is ready and waiting whenever you have time). there are probably ways to improve this like a pump and jet to stir things up.

If you can get your pool whipped into a something that will be a good home for fish I have some other ideas I think could help.