I've done significant reading on pond clearing. The big thing is to pull a few decent sized samples, quart mason jars should theoretically be enough.
Let the first one sit to see if it clarifies on its own. If that's the case, the big issue may be that you have fish and critters stirring things up in the pond. The only way to fix this is to introduce predators that can knock out the population of the mud stirrers or to initiate a mass pond die-off, there are "non-persistent" chemicals that can accomplish this, then after your water parameters come back down to normal after everything has rotted down, you can work to start rebuilding the pond life with animals that are less likely to stir mud, however there's no promise you won't get transfer again.
If the samples don't clarify on their own, you can try adding various amounts of gypsum, barley straw, etc.
Also, aeration via an air pump may help significantly if you have a good way to run power to the pond, but I haven't found anything particularly affordable to execute this, interested to learn if you find something.
If all you really want is for it to look less gross, and the idea of putting food grade dye in your pond doesn't upset you too much, pond dye is a legit strategy. It also helps reduce algal activity by helping to block sunlight from penetrating very far into the water. For our .66 acre pond, last year we used a gallon of blue dye and half a gallon of black dye and were pretty happy with how it looked. This year we did a gallon and a half of black dye and found the results to be less visually pleasing, I think next year we'll probably do a gallon of black and half a gallon of blue.
We had a significant amount of erosion present around our pond as a result of the cattle that used to run the land, and the tree cover around the back portion of it. I have succeeded in increasing the grass cover on the dam portion and banks of the pond, however we do still have game trails coming to it, and the treed area is dense enough there's nothing but trees growing. I hope to start thinning the treed area at some point, but I have so many other competing priorities that it'll likely be several years before I get really serious about that undertaking. I have invested in some blue flag irises, pickerel rush, etc for the margins, but I'm unsure if it died or has just been overtaken by the grass I've been encouraging on the pond dam. I also introduced some triploid grass carp last fall to help eat the mat of floating pond weed we have thriving around the edge, but from what I've read, they take 2-3 years before we will start seeing any type of impact, and at that point it'll be time to go buy more to help restock, so who knows how that will end up shaking out.
Here are a couple of blog posts I've written about our pond's appearance.
https://thebluegrasschapter.wordpress.com/2021/09/24/a-look-at-the-pond-over-the-course-of-our-time-on-the-land/
https://thebluegrasschapter.wordpress.com/2021/05/31/pond-dye/
https://thebluegrasschapter.wordpress.com/2021/06/04/pond-dye-update/