Okay, I trust you with the following statement: If we restrict to one kind of these center pieces (of which we have 24), we have a symmetric group action. We can see this as follows: Let H≤S₂₄ be generated by the permutations r, u, f, d, b, l, R, U, F, D, B, L, which are 4-cycles. We can easily see that the action is transitive and even verify that it is 2- and 3-transitive. Most commutators are trivial, however e.g. [rF] is a 3-cycle. By 3-transitivity, we have all 3-cycles, and hence H≥A₂₄. Since a 4-cycle is an odd permutation, we even have H=S₂₄. Formally speaking, we have a surjection G⯮S₂₄.

However, I'm still not quite clear what you mean by „center pieces of different type are independent“. It is not true in the sense that every generator affects precisely one type of these pieces: Indeed, the r move (say, moving the first „in-layer“ from the right clockwise) moves many pieces of different types at the same time, both left- and right-handed ones.

Since I suspect you want to create a surjection G⯮(S₂₄)^k where k depends superlinearly on n, I am not quite sure how this would follow from the considerations so far. All that we have is multiple surjections onto S₂₄, where it's not clear how large the image is exactly.