I had these same concerns recently, since don't have a Sidewinder system, and I couldn't figure out an ideal efficient packing with my stove in the conical sleeve and my pot set. I also noticed that using the system with wood leaves a bit of soot, and rolling the stove walls every day for storage was bound to generate a lot of mess on my hands to clean up.
I instead cut out a sleeve from 2 free Tyvek mailing envelopes, sealing the edges with superglue. As shown, there are two main sleeves, with a strap looping over the smaller to keep them together, and to provide a handle to pull the stove out of the pack when desired. Hidden is a small loop under the lip to secure the stakes, and a separate sleeve for the mesh strip. It weighs barely over 1/2 oz.

Here's the whole kit emptied out.

Using the kit this weekend was very easy; the package was easy to slide into my pack even when full, against my tent (or any other large soft item) and the rear pack wall. It was also easy to remove from the pack without removing any other item; the Tyvek slides easily.
The one caveat is that I noticed a small nick in the corner of one of the dovetails. I don't know if that is from this storage method or not, but in theory the dovetails are a bit more exposed to external impact or human negligence when flat. I plan on molding small dovetail protectors out of Instamorph / polycaprolactone for protection before my next trip.
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