In general the critters you see might be called water fleas, but that's just a catch all. If you're interested in an exact id you could
start here. Once you know what you have and understand their life cycle you can make an informed choice. In general they consume stuff you don't want in the pond and in turn provide great food for fry or larger insects that bigger fish eat. I know in this forum the overwhelming opinion is that muck is toxic, but it does support a vast and interesting ecosystem that some people enjoy.
As you have seen Koi can certainly thrive in those conditions if not too many for the pond size, or fed more than the pond can break down. But many here would strongly disagree. Ponds can range from show Koi pond, to water garden, to wildlife pond. All have different attributes.
As for water clarity...I've cleaned a lot of my ponds in my day and after awhile the term "green" wasn't enough to tell what the problem is. My guess is if you look really close you will see tiny suspended flecks, kind of like ground pepper, only pale. I guessed that because you said "filthy quick".
I'm assuming your pond has been around for more than a year. If true the "green" you see could be colonies (clumps) of single cell algae. They clump to protect themselves from the sun's UV. The clumps trend to sink in still water or over a long period. Stirring up the pond can get these suspended again and added to the always present single cell algae the water appears green, but a darker green. In new ponds, with new algae blooms there are only single cell algae, no clumps, and the water is bright green, almost lime green, but it depends on the species.
I've also assumed when you cleaned your pond you didn't empty and replace all the water. That could get you the bright green new bloom.
The reason I say all this...fabric filters can be effective removing suspended flecks and, depending on the fabric, can even remove algae colonies. These can clear a pond over night. On the other hand a UV won't remove the flecks and have a limited affect on algae colonies but will kill them. The flecks also block the UV making it less effective. On yet another hand the UV is very effective on new bright green algae and can clear a pond in a few days.
In all cases the UV has to be installed correctly, correct flow, etc... This is especially true when the water has suspended flecks.
When the goal is crystal clear water my strategy is to clean the pond. I don't know how you cleaned yours, I used
vacuums. The key is to repeat after a few days, or a week. I start by vacuuming large stuff, leaves, sticks because they clog the next step. That stirs up the muck and I wait a few days for it to settle. Then vacuum with something that removes water, only way to get out the fine stuff. Wait a few days, repeat silt vacuum. Repeat until you're not getting much.
Now the fabric filter. Another choice is a bog filter which allows the flecks to settle out and can grow heterotrophic bacteria that kills single cell algae with antibiotics.
The reason for all the vacuuming is the muck will be a source for more flecks. The flecks are just decomposed organic that have broken down to the point of becoming suspended. Muck is the compost pile. So, if you just use the fabric filter the muck will throw off more flecks and you'll see a limited improvement.
Also, the stir during the vacuuming can help speed up decomposition. That with less stuff to decompose can help a bit.
You can also add high end filters specifically designed for each item in the water.
The most common choice however is to add some little combo filter deal. They can help. But in general the water will clear on its own so the filter gets the credit. What I outlined always works.