This question kind of brings up some more fundamental questions about fermentation (like whether or not you could even call something kombucha if it's not made with a 'proper' SCOBY, given that a SCOBY could just be said to be 'some collection of microbes that, together, produces kombucha'). It's a bit of a chicken-and-egg situation.
For most cultures (including SCOBYs, sourdough starters and koji [the 'mouldy rice' you mention]) the real idea is to make a 'selective environment' (that is, an environment which encourages growth of wanted microbes and hinders growth of other, unwanted ones). What makes it great is that these selective environments can also be tasty things to eat and drink (kombucha, bread, miso).
All cultures arise originally as random encounters of wild microbes with environments suitable for them to grown in. Imagine, a few cells of yeast floating on some dust, landing in an open bowl of grape juice... wine! People long ago discovered 'culturing,' that is, using that first, 'accidental' wine to start another wine fermentation with fresh grape juice (rather than just hoping for another chance encounter). This reuse, continued generation-after-generation, eventually leads to the selection of the best community of microbes for the job (people being more likely to reuse a culture from something they liked, rather than something they didn't).
For sourdough and koji, the starters are easy to make from scratch because there are well-known natural sources for, and fairly simple methods for growing, the appropriate microbes (the flour in sourdough comes from grains [wheat/rye/etc.] which already naturally host many of the right microbes on their surfaces; the proper molds for koji are known to grow on corn husks, and so these are sometimes used to inoculate rice). Knowing this, it's easy to start and perpetuate the proper selective environment for their growth.
Kombucha SCOBYs are a special case: for one, the substrate for kombucha (tea, sugar, water) contains none of the appropriate microbes (unlike sourdough) for seeding the culture, meaning it is entirely reliant on the SCOBY for these.
Second, they tend to include more microbial diversity, each species having its own particular nutritional and environmental requirements. For example, it's important that the SCOBY floats, since acetic acid bacteria require contact with oxygen. It's possible that the part of the SCOBY that is exposed to air has a different microbial make-up than the parts that are submerged. Also, there are complex interactions between the microbes and their byproducts (for example, in kombucha, the acetic acid bacteria need yeast to first convert sugars in the liquid to alcohol, which then can be oxidized to acetic acid).
Lastly, the SCOBY is a solid mass that floats in the substrate, while the other cultures mentioned fully mix with or permeate their substrate. Because the microbes are basically fixed in place, the overall culture changes and adapts much more slowly. Only the microbes on the surface of the mass can really enter the liquid, where they can move and grow more freely. Whatever microbes are dispersed in the liquid (which of course came from the surface layer) are those that then gets 'glued' to the outside of the growing SCOBY (by cellulose or similar stuff generated by the microbes), becoming the new surface layer.
So, you can probably start to see that the dynamics of how a kombucha SCOBY adapts to its environment are different from many other cultures.
The take-away, I suppose, is that it just takes much longer and is much more fussy to develop a SCOBY with the proper characteristics (i.e. that makes good kombucha) than you might expect it would be to make a healthy sourdough starter or koji culture. Of course it's possible to develop from scratch (all of the required microbes are readily available to the motivated wild-fermentationist). I've heard of people using vinegar-mother (acetic acid bacteria culture) to start a new SCOBY, to which could easily be added cultures of wild yeast and other microbes (or just hope they happen to float in). Or you could take another fermenting culture (from sauerkraut or pickles, say, which are rich in yeast and lactic acid bacteria) and simply feed it on sugar and tea and treat it like kombucha during fermentation, hoping that any acetic acid bacteria thrive in a change of environment. Hypothetically this could eventually yield a recognizable SCOBY.
I don't know if this really answers your question (it probably just brings up more), but I hope it at least helps to give an idea of how and why kombucha SCOBYs behave differently than other cultures and why it's probably been hard to find advice on starting one from scratch.