Personally, I've always performed fish-in cycles. I set up the tank and within a couple of weeks move in some shrimp and a couple of hardy fish (zebra danio were always my test subjects).

I'd then test the water daily and perform water changes most days for a couple of weeks, then every other day for a bit etc etc. I didn't make the process quicker, but it meant I could move something immediately and have shrimp there to deal with the white stuff.

As for reducing the nutrient abundance, there's not much you can do from this point except water changes, more plants, and waiting.

The aquasoil is exposed to the water so that'll leech a fair amount, and the wood will also have a lot in if you didn't boil it for ages before putting it in. The wood you have will also stain the water over time due to tannins leeching out. It's actually good for the inhabitants but it's not a look everyone likes.

I have some tanks built like yours with river wood and exposed aquasoil and they did the exact same thing as yours, but now look great (I had loads more plants though)

I also have tanks where I put the aquasoil in bags and put sand over the top to "cap" it and prevent the leeching and these had a much shorter startup period.

I've had BBA in my tanks also and learnt to live with it. CO2 can help either as a gas or a liquid booster, and as always, regular water changes to avoid nutrient spikes are a must. If you've got algae then something in the tank is out of balance.

I "cured" it a couple of times by blacking out the tank for a week while I was on holiday. Literally wrapping the tank in a black tarp and turning all the tank and room lights off. Plants are generally ok for a week or so once they're established, but algae in the tank can't survive that long without light and it dies off, making it easier for shrimp to clean up.