Just an FYI- those are logs, not stumps. Stumps are the bit still attached to the roots in the ground. The fungal growth is perfectly normal and natural for wood left exposed to the elements. Health hazard? I don't know, but I climbed all over log piles like that as a child and I'm fine.

The tree trunks have been cut to fireplace lengths, so your problem is much easier than if you had tree stumps in the ground that had to be dug out.

Depending on how much you want to spend and what is available, you can rent a chipper shredder and grind the pieces into mulch. From the picture I looks like the chipper shredder needs to be a large one. You could get by with a smaller one if the log can be split in to smaller pieces.

Worse case scenario you may need to get some one to carry them away. If you are lucky you may find someone who will do it for the wood, probably you will have to hire some one.

To my knowledge the fungus growing on them would not be a threat to the children.

If you like to watch the wild life in your yard the fungus will attract many different animals Deer, etc.

Our 'Solid Waste' department will take any kind of organic garden/yard material and tree wood of any kind. Generally, you have to haul it or have it hauled there but there's no charge for them to take it. They may even pick it up at your place.

You might want to call yours and get the specifics.

Where I live, all you would have to do is carry them to the side of the road, and put a FREE sign on them. They'd be gone within 24 hours.

If that is too much work, advertise 'FREE FIREWOOD' somewhere like NextDoor.

My guess is it's Ash, killed by the emerald ash beetle. So long as the wood doesn't get hauled more than a few miles, I wouldn't worry about it. The beetles are already there.

Rotting doesn't sound much good for firewood. :-(

If you don’t plan to burn the wood in a fireplace/firepit or use it for mulch, consider having it hauled away.

The wood looks like it has been there for awhile. Before your child crawls on or near it, check it for ants, termites, bees, snakes, mice, etc. In my area, an old wood pile will attract all of these.

it doesn't take many years for logs to rot. When they start getting spongey just finish them off and spread like mulch. My recommendation therefore is do nothing.

The kind of wood determines how long it takes for it to rot to the point it can be broken down, and the biological activity in the ground it is on. Around my lot, pine will rot away in several years. Oak, cypress, cedar, take many more years, some time decades.

Moss, mushrooms, molds and fungus are everywhere outdoors, including on logs, dirt, and anywhere there is organic material.

Decaying wood has its perks in the yard. I used the dead trunks as partitions/boarders in the yard after a hurricane. Noticed a massive and magical season of lightening bugs the first few seasons. And something to be said for exposing kids to nature and not raising them in an antiseptic environment.

You don't want your toddler to eat the mushrooms, although most of the mushrooms I see on the logs are shelf mushrooms, which tend to be woody. They are generally listed as inedible (as opposed to poisonous) because they are as appetizing to eat as cardboard or fiberboard, so people don't eat them. I'm pretty sure none are poisonous, athough some may cause some mild intestinal upset. You can also find jellies on rotting wood, some of it quite pretty. Most are actually edible, albeit not always all that tasty. AFAIK, none are poisonous. Then, you can get coral mushrooms. Some have a laxative effect, but none, AFAIK are poisonous. However, in general, a toddler who puts anything in his mouth, is going to spit out any mushroom he picks, in the unlikely event he sticks it in his mouth.