I was thinking about this, actually... I think it's possible. I haven't fleshed out the proof, but here's some intuition:
If we can show that we can move the corners, edges, wings, and centers however we want (independent of the other pieces, that is), then that should be enough. Corners and edges are given, since there isn't any restriction on <F,B,L,R,U,D>. Wings can be shown to be given because it's definitely possible to solve wings relative to their midges without <Bw,Lw,Dw> - just do beginners-style 3-cycles over and over, flipping edges around using rotations and R U R' F R' F' R.
The final thing I want to prove is centers - intuitively, we can use a heavily-modified version of U2 to always push the targets to a place that the U2 alg is in our set (eg, the location Fru, which we can do by f' u f U2 f u' f' - so, moving a piece like Bdl could be done with B2 u2 [Fru alg] u2 B2). Not super sure how to handle niklas-like cases yet, but it shouldn't be hard.
Obviously, it's not gonna be move-efficient, and a random-move 4x4 scrambler won't have a reasonable distribution of 5x5 states, but that's not the question at hand
Remember that you can reach all 3x3 states without ever doing a D move! You can just do L R F2 B2 U L R F2 B2 every time you would normally get one