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I thought about mixing some of the sawdust, glue, and stain to make a filler for the gouges.
I've seen this advertised as a solution many times.
IME it does not work. At least to my eye.
The sawdust is all end-grain and therefore makes the "patch" absorb stain more readily.
It leaves a uniform, dark colored spot in place.
It's just a sliver, but a patch of solid wood can be glued down into that gouge and would be more invisible to the eye.
Personally, I'd research how to clean off the art residue ... give it light sanding, and apply a good topcoat.
For me, the dings, gouges and such that my children have applied to our kitchen table are part and parcel of their lives.
I enjoy the memories.