It's that word "natural" that's the problem. Since (as asserted above) God is not "natural," scientists doing science must assume that He plays a role in any natural process. Conveniently for the atheists, this approach guarantees that the scientific method must place itself in opposition to the possibility that God exists. It also guarantees that there is no possible "scientific" test that can demonstrate the action of God.
The logical system derived from this belief is useful. This does not prove the premise's truth, but it proves the usefulness of the premise.
The necessary and sufficient condition for the logical system is that phenomena must be explainable in terms of consistent mechanisms. There is no logical need for the additional proposition that God does not exist. The roots of that assertion rest in something other than scientific necessity.
1) Observed phenomena have natural causes.
2) Time does not change explainations for observed phenomena.
This is a simpler set of premises than including God. I don't think that adding God adds anything to the scientific explanation.