Yes, mycorrhizal fungi, I googled-images, this one on top of a page is perfect match: https://fungi.com/blogs/articles/get-associated-with-mycorrhizae
From my past huge experience growing cactuses from seeds: my first attempts were to sterilize everything, including boiling soil in a water (no microwaves were available at that time); putting seeds to very dark potassium permanganate solution for 1-2 days. By accident (by laziness) I found germination rate and even growth speed were amazing with just peat moss mixed with sand. Recently, I was trying new recipe: SumiSoil from https://www.leevalley.com/en-ca/shop/garden/supplies/soil/59368-sumisoil which is Japanese ceramic-coated charcoal, 100% sterile (almost), and charcoal has anti-bacterial properties. Although plants (cactuses: ariocarpus, astrophytum, etc.) were absolutely healthy and germination rate was over 90%, they were growing at least 10x slower than seeds in regular unsterilized Cactus soil mix from retail store.
Potassium permanganate helps: seeds get washed, sugar from surface removed, I had much less nasty creatures; and I used the same light-rose solution to bottom-soak seedling pots until I see moisture on surface: only bottom-soak, so that soil doesn’t get compressed. I noticed it doesn’t really has impact on beneficial fungi: I had the same tiny white threads helping sprouted seedling to get upright instantly.
I tried another solution for cactus seeds found recently on internet: to use (sterilized, washed) sand in styrofoam cup, without holes in a cup, put water till almost top, put seeds in a water, wait till water dissipate and seeds are on top surface (sand), put in zip-lock, etc. Yes, it worked, but seeds were in centre of cup… germinated, but growth rate was very slow in comparison… fertilizers didn’t help. But benefit was: very easy to separate seedlings and replant it in regular soil without any damage to roots.
Nothing worked as good as non-sterilized soil from retail store, and pre-soaking in potassium permanganate (which is indeed “potassium”). Or without pre-soaking, but still using rose-colour solution for bottom-soaking pots with seeds.
I can see some benefits of stratification: if you are not in a rush, and you have rare seeds with germination spawn over time, you can pre-germinate in a fridge and then plant one single plant in one single pot (or directly in backyard).
P.S. My experiment with seeds above doesn’t include any potassium permanganate; soil was from retail store, ProMix, mycorrhizal fungi in it; I only heavily used 3% hydrogen peroxide on Morus Nigra seeds daily, first two weeks, to kill some brown-colour fungi. I think I didn’t kill fungi; just washed seed shells from sugars.
P.P.S. Forgot to add about importance of UV light: when I just started with cactuses, I read that UV light has huge benefits (killing bacteria, fungal infections, etc); and temperature shifts are important for germination; and that fluorescent bulbs generate UV 
so that I use only “bad” fluorescent bulbs (from LeeValley), and plastic boxes / zip-locks instead of glass (because plastic is transparent to UV). 30-35 degrees Celsius during day, and 20C at night - best for cactuses; but in my environment for Mulberry it is probably 25C-27C during day (ziplock gets heated from light above; it is really like warming sunlight, don’t try to save on watts-efficiency) and 21C at night.
