would mean a virtual end, ... an avoidance of... , and ...

We have to continue the parallelism by using a noun form. This eliminates options A and B, which use reducing as a verb.

C. "as they do often" is incorrect, it should be "as they often do". Also, "a reduction of" is unidiomatic, it should be "a reduction in"

D. Changes the meaning by suggesting that the reliance was on the hidden amounts of sodium and not the processed foods themselves.

E. This is correct. It uses the noun form "reduced reliance". "..that contain..." correctly modifies processed foods, and "significant but often hidden amounts" is acceptable.