Not necessarily, but many scholars like the American author, Robert Louis Stevenson, believe that L2 writers should use reading as a vehicle for strengthening writing. However, in the act of reading, one should be a "diligent ape" noticing the styles of native speakers and using original stylistic particulars in their writing until their own personal writing comes into focus and is consolidated. Your analogy envisaging writing as a ladder is insightful, and to me, it is the most practical way of learning to write in L2.
I believe that reading helps (particularly extensive reading). However, improvement occurs unconsciously; that is, structure, vocabulary, discourse pattern, ... would be acquired (not learned).
In the writing classes I teach to native English speakers, I find that the best writers are almost always those who read extensively in their younger years. Reading gives people a natural feel for how to put words together into engaging narratives. As Sarah says, this is mostly an unconscious process, and it does not happen quickly.