There are tons of fungal spores everywhere. You want to maintain the local fungal community without dragging in out of locality fungus. There will be less fungal spores on the forest floor than there will be in your out of doors garden. Forests and ecosystems are so efficient you won't find anything going to waste. Certainly lots of organics in all stages of decomposition, but as soon as something has been decomposed it will be eaten and taken into the soil, pooped out actually. But to find chemistry such as N P K not going to happen unless something is very wrong with the ecosystem.
Fugal spores, bacteria...decomposers especially are in the air, on your hands...everywhere. You do not need to ever worry about inoculating your soils with more. That can be a very bad thing. To add exotic fungal spores, bacteria to soils without beneficials to keep in check is a very bad idea.
Bark will have fungus on it, bacteria...decomposers. Because bark is not decomposed. I would throw nitrogen on your compost to feed those decomposers and make their work go faster. There will not be any nitrogen left to call that compost a fertilizer of any sort.
Instead of thinking about moving exotic fungus and bacteria into the garden I like to imagine making the life I've already got in the environment of my garden more healthy and better able to reproduce. A little fungus, balanced with checks in your soil is more than enough. A little more? Not such a good thing.
Mycorrhizae doesn't...ummmm...transfer chemistry to the roots from the soil. It 'assists' but the plant is not dependent on the mycorrhizae to grow. All plants are different have their own type of mycorrhizal buddies. Fungus is everywhere.
Adding mycorrhizae is helpful when you are transplanting mature plants into subsoil after major damage, destruction during a construction project. There would not be an abundance of help or hope in that situation, mycorrhizae has done miracles. Otherwise it is everywhere in normal top soils. Local mycorrhizae already established. Just needs to be...fed. Just organic matter...a little is all that is necessary. The web of life is humongous and amazingly complex in soils that have had stability within an ecosystem.
Just keeping our soil healthy and adding some decomposed organic matter, using cover crops, incorporating and enhancing the drainage by raising plant beds, adding a little fertilizer that one actually knows the numbers...you will have plenty of life in your soil.
How do plants in sterile potting soil do so well? Plants in hydroponics? No mycorrhizae right? Hadn't even considered hydroponics and symbiotic fungus until your question. Good one.