The edges of the chamber don't usually burn in my pipes, as the way pipes work is for the cherry to mostly stay centered and the flavor comes from the heating of the surrounding tobaccos. This is similar to how a cigar burns, with the cherry on the cigar burning more deeply into the stogie than the surrounding tobacco. I usually just push the edges back to center with the tamp as the bowl progresses, but that is not absolutely necessary.
The "wads" of tobacco are parts of a flake. The blend was designed like that to slow the burning of whatever flake that is, so that there is a certain "experience" with flavor that the tobacconist has designed for you. Burning all of the tobacco in the bowl is absolutely not a necessity. The "stems" you pointed out are actually the veins of the leaf, and probably not the ribs. The ribs tend to be fairly large. Some blends may have a few ribs in the mix, because their is more nicotine in the ribs and stems.
IMO, you're fretting over something that most of us don't even think about. It's just a part of the pipe experience. There are actually very few blends that were meant to be burned to "fine white ash" as we pipers say it. These are blends where the leaf was cut specifically for that task.
Don't worry about burning the bowl from edge to edge in the chamber. If you do, you will get less flavors and most likely increase your chance of getting tongue bite. And, don't fret about having bits and pieces of unburned tobacco in the bowl. Some of us will even dump the dottle... see we have a word for the unburned, "dottle," so if we have a word for it, we are ok with it. Ha ha.
Just enjoy the flavors, dump the bowl when you get tired of it, and don't sweat it. Just enjoy... :
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However, if you just can't enjoy smoking with these slight imperfections you might try a cigar, as they burn way more cleanly through the entire leaf because of the way they are rolled.