Having previously owned a chlorine pool, no matter the issue, there was little that adding shock to the pool wouldn't cure...and fast. The biggest takeaway from an almost full season year with a natural pool is that, like nature, it likes to take its time.
As temperatures change and seasons roll around, you adapt as the pool adapts. When the early spring starts to bring the plants back from winter hibernation, just about the only maintenance is washing out the aggregate filter from anything that made its way in during the winter (we keep the water running all winter since our winters are generally pretty mild with temperature rarely dipping below 0° on the coldest nights).
Just about the only thing that's similar to maintenance of a regular pool, is the cleaning out of return baskets from topwater debris. We do have a Maytronics Dolphin Bio robot and use it fairly often.
We've had success keeping the pool in balance (clear water, low to no algae growth) by using Mick's blue/black dye to prevent full sun from growing algae faster than the plants can consume it, and applying beneficial bacteria to help accelerate the consumption of any algae that does begin to show up. There is a thin film of algae that covers the plaster that is slick, and gets slicker as temperatures hover in the mid-90's during our summers. We built really large stepping 'pads' into the pool, but caution is always given to guests to watch their step entering and exiting. As the plants begin to grow and multiply, they take over the heavy lifting of producing clear, clean water that's drawn from below the roots and circulated back into the pool. The key to all of this working like a clear-running creek (which is essentially what you're trying to replicate) is pumping enough water to turn over the pool's total volume, three times in a 24 hour period.
If you don't think twice about jumping into a lake, this is the same experience, only with much cleaner, clearer water. Seeing the robot nine feet down perfectly clear is not something you'll find in a lake or pond. And all it really takes is electricity to run the pump.
But, just like a lake or pond, critters come with the territory. Mostly in the form of water bugs (little rower bugs), snails and other algae eaters barely visible to the eye. The less algae (better control you have of it) the less critters. The top of the food chain in our pool is a slider turtle that had grown from a half-dollar sized shell in an aquarium to about 10" around today. He absolutely loves it. Then there are the frogs. Not that you ever really see them because they hang out in the thickest of the plants, but you can certainly hear them, particularly in the early amorous springtime, right after dusk and into the early morning.
The sounds around the pool (water splashing, frogs chirping) and sights (dragonflies flitting from lily pad flower to the next and birds splashing about) are some of my very favorite reasons for a natural pool. It's far more than just a swimming pool, it's a habitat and oasis that feels just right on our small acreage 'farm.'