Disclaimer: I have not attempted this so I can only approach it from a theoretical perspective.
One of the biggest issues I can see with a "fully natural filtration approach" is that the phosphorus cycle doesn't work too well in reef tanks long-term. Carbonate surfaces do bind phosphate, but they dissolve most of it back once the water's phosphate levels drop again, so the rock and sand buffer phosphate rather than permanently removing it. That leaves water changes, harvesting macro algae, and a skimmer physically taking out particulate matter that contains phosphorus as the only means to export phosphorus (besides permanent growth like corals which is slower than the phosphate import).
Without a skimmer you will also reduce gas exchange, which is exactly the part of the carbon cycle that gets rid of carbon for you. You are limiting how much carbon dioxide can off-gas, and I can imagine you will notice this with lower pH readings sooner or later. We don't know too much about the organics in our tanks, but there is some evidence they play a role in cyanobacteria outbreaks, so it is in your best interest to have a decent carbon export, even if you are also carbon dosing to boost denitrification in the sand bed like Lasse does.
Additionally, all biological matter has to be fully broken down before the bound nutrients can exit the tank through the carbon and nitrogen cycle. A skimmer would remove particulate matter (containing C, N, and P) before those nutrients dissolve and become detectable by your tests.
"Biodiversity" is one of those terms that gets thrown around a lot, but as far as I can tell from AquaBiomics' observations it starts low in new tanks, builds over the first year or two as a tank establishes, and then older tanks tend to test lower again if you don't actively do something against it. You would need to continuously add live rock, sand, and mud to maintain biodiversity. Alternatively we might accept that biodiversity in a tank sounds really nice, but maybe it's not a requirement for a healthy tank as long as "the right organisms" have established. Otherwise tanks like mine that were started from 100% dry rock and sand, as well as very old tanks that haven't added any new live sources for years would struggle.
If you still want to try it, I would limit livestock to a minimum and get an absolutely oversized filter sump.
Edit: I'm not saying it's not possible, but it might prove challenging.