I am going to try to put what the OP is claiming to try to do into a mathematical framework.

Let S be the set of positions that can be solved in 10 or fewer moves. (The value of 10 here is rather arbitrary, but I'll just use this value for this discussion.)

Let G be some subset of the elements of S. (I chose G for "goal". The goal of the first phase is to reach an element of G.) These elements may be chosen based upon some properties of the positions, such as some relationship having to do with the permutation and orientation of the pieces.

Let G[sub]N[/sub] be the set of elements in G that are optimally solved in N moves. (That is, we partition the set G based upon the number of moves required to solve each position.)

We want to find a "method" such that we can show that for every position of the cube, we can reach some position in G[sub]N[/sub], for some N, using no more than 20-N moves.

(And not only this, but we also want this method to be human learnable.)

There are two big obstacles with this.

1. Choosing a set G small enough to be human learnable (the human must know how to optimally solve every element of G). But G must also be large enough to make it possible to accomplish the first phase in few enough moves.

2. Figuring out how to reach an element of G from an arbitrary position in few enough moves. If we ignore the "human learnable" requirement, we "simply" need to show that this is possible.

I think trying to make (2) a manageable size problem is generally going to force increasing the number of elements in G, making the first obstacle worse. Likewise, trying to make the first obstacle manageable is going to tend to make the 2nd obstacle all the more difficult, if not impossible.

In the end, I assume some concessions will need to be made. These could include:

1. Reduce the number of positions that must be solved in 20 moves. If, say, we only require that only 99% of the positions need to be solved in 20 moves, we remove the requirement that all of the millions of 20f* positions need to be solved optimally.

2. Require the average move count to be 20, rather than 20 as the worst case.

3. Relax the move count that is required to be achieved to some larger number (21 or 22 or 23, etc.).