In no particular order here are my thoughts on the various issues:

Fungus gnats are probably there in outdoor plants too but in much smaller numbers - too much competition, predators, and harsher conditions for them. Indoors, it becomes more apparent since it is much more comfy for them.

Fungus gnats feed on composting matter and live weak tissues. I do not think peat moss is a great food for them but it certainly holds moisture long enough to decompose leaves and twigs that make their way to the soil. That is why keeping your soil surface and surrounding area free of plant matter is a good thing indoors.

Perhaps bagged peat contains or contaminated by decomposing plant materials and that could be the source for fungus gnat problems. Although peat is shipped so dry I do not think anything survives in there.

Probably neem cake will work - but it is also adding food material to soil. And frequently adding neem cake will just make a free draining mix like 511 clog up. I see no harm in trying it a couple of times though.

If you can have the plants outdoors I would do that first before repotting. Let it get used to the outdoors first. Fungus gnats will diminish. And when it gets warm enough do a full repot. Keeping the old soil in the root ball is asking for trouble if you are going repot into 511. Not only all the pests will still be there the plant will suffer being two very different soil mix. After you repot, keep it in shade and protected from wind. You also need a stake to keep the plant anchored well while it the new roots are forming. After a couple of months the stake can be removed.

In winter, if we get a rainy day with 50F+ temp then I take some of my plants more prone to pests (aphids, mites, scales, etc) outdoors. Does wonders for the plants.