The basic ideas of group theory are definitely accessible to grade schoolers, and even the fancier parts of an undergraduate group theory course (Sylow theorems, etc.) are accessible to high school students who go through the work.
I wouldn’t hesitate to recommend Nathan Carter’s lovely book Visual Group Theory to any bright 15 year old. But for a (say) 10-year-old, just learning about the symmetries of shapes and tilings is plenty of useful and interesting math.
The Sylow theorems are by no means “fancy”. If you don't understand them, then you don't know group theory, plain and simple.
Now, we can debate the merits of exposing children to groups before they can understand group theory. But learning group theory means learning (0) the axiomatic definitions of group, subgroup, coset, etc. (1) the main theorems in group theory and their proofs.
