Neighbor's tree - Security and dirt. Who is responsible?

So I don't see any danger for your child here. At least as long as it stays with the small branches that can be seen in the pictures. Yes, the tree makes a mess, but it was there first. I can understand your neighbor to some extent that he is not exactly thrilled when suddenly the new neighbor arrives and wants the trees to be trimmed. Maybe you should let things settle for now and revisit the conversation in the fall, offering to cover the costs to have the branches cut back on your side.

If I am seeing it correctly in the pictures, they are coniferous trees, which is also supported by the cones. The branches are completely dead except for the crown area. I could imagine that in a few years none of the trees will be alive anymore and will then have to be removed anyway.

The fact that the owner refuses removal might also be related to the fact that the trees stand between the boundary and the shed or whatever that may be and cannot be easily removed.

Just suggest to him that he lets them be removed in autumn. He could then let the trees fall onto your property and then cut them up and transport them away there.

You can also mention that you plan to build a tool shed in the back part of the garden or to plant shrubs, trees, with appropriate spacing, which means that a simple and thus probably also cost-effective removal for him will no longer be possible.

Maybe this will lead him to take action.

Section 910 of the Building Code, paragraph 1 applies here. Set a reasonable deadline.
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It would be nice if it were that simple :)
Because there is also paragraph 2 of Section 910 of the Building Code:
"(2) The owner does not have this right if the roots or branches do not impair the use of the property."

I generally do not see the use of the property impaired here. If the owner has a different opinion and the "tree owner" stands firm, despite the Building Code, nothing is gained. Thus, probably only "imminent danger" due to sick trees would remain. But that can also become complicated, and Section 910 of the Building Code says nothing about that.

I basically do not see the use of the property impaired here.

Just look at the soil. Due to the needles, the soil is completely screwed (acidic). That will not get any better. The impairment is significant.
But,

If the owner is of a different opinion and the "tree owner" plays stubborn

, the only option left is an expert assessment. This is the usual way.

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The impairment is significant.
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Significant? I was talking about the fundamental suitability. The plot does not become unusable at the location of the trees.

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Because of the needles, the soil is totally screwed (acidic).
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In the first picture, it is, in my opinion, visible that the entire (visible) soil is "screwed." Besides, the garbage dump by the fence would be a bigger (visual) nuisance to me.

For a toddler, I see - exaggeratedly - the stacked pallets as a greater hazard. The child could climb on them, jump off ... land awkwardly, break a foot and neck.

Just look at the ground. Because of the needles, the ground is completely screwed (acidic). It will not get better either. The impairment is significant.

The problem is unique. If I infer the condition of the 4 trees from the state of Teuto and Wiehengebirge, 50% of them are dead and not producing needles. The rest will inherit the bark beetles this year and die as well.

Then it would be time for felling, as they will break or snap in the next storm. However, the OP does not need to worry about the children because they should not be in the garden during a storm anyway.