Fungus is actually very desirable, but it does depend on the fungus in particular. Most are harmless and saprophytic, so they break dead organic material down and it results in plant available nutrients. Symbiotic fungi such as mycorrhizae are quite invaluable because these are fungi which practically live solely to feed plant roots and keep them happy. The presence of harmless\beneficial fungi also prevents the real nasty kinds of parasitic fungi from taking hold as odds are they will already be out-competed for food and space.
I personally just wouldn't start seeds in it directly due to the possibility of opportunistic fungi rotting the seeds (more likely to happen under compounded adverse conditions, but still).