The topic of strops and using them... It's easy to get caught up in almost religious discussions about it.
I presonnaly use linen and leather, because it makes sense to me to do so. It makes sense because it fits into a physical theory about stropping that I believe to be sound. That theory is based upon a number of scientific articles about the conditioning of steel surfaces, studies about knife sharpening, and personal observations. I have described that theory in the article about stropping, with the disclaimer that hard evidence is only partially available. The rest is hypothesis, which is a scientific euphemism for speculation. But still, I vastly prefer speculation based on logic over religious believes, hence I stick with my theory for as long as new fact and insights don't force me to alter it.
Based upon that theory:
Linen has a cleaning function, and it excels at that over leather.
Leather has an aligning function, and it excels at that over linen.
Allow me to elaborate:
When freshly honed, an edge has all kinds of sub-microscopic debris sticked to it. We can't see it with an optical microscope, but a scanning electron microscope can. Most of that debris comes from the honing process, but there might be some corrosion present as well. After all, we are using water in the sharpening process.
After shaving with a razor, regardless how well you cleaned the edge, sub-microscopic corrosion will be formed at the edge, unless you are as uptight about oiling as I am, but even then, there might be sub-microscopic pieces of steel present, partially teared of the edge during the impact with coarse beard hairs.
All that stuff needs to be removed.
You could do that on leather, but I don't think leather is the safest and most efficient choice. I say safest, because whatever comes off the edge, will certainly stick to the strop. Iron oxide has a few different forms, with one constant: it's harder than steel. We may call it "rust", jewelers like to call it "rouge" and they use if for polishing. But the kind of oxides that end up sticking on the surface of our leather strop, do not have a uniform particle size. Some of these grains may be quite coarse. And I believe they are the reason why well-used leather strops can cause stray scratches on the bevel, running all the way to the very edge...
Linen, being a woven fabric, is far better at accepting this debris and embedding it safely within the recesses of the fiber. (nonetheless, it does need periodic cleaning). A decent linen (I'm not talking about that horrible nylon seatbelt stuff), does not leave those stray scratches, and neither does clean leather. That is a statement I have personally confirmed.
So, basically, a first function for the linen is to keep the leather strop clean. I also have strong indications that a good linen is more efficient in debris removal than a leather strop. Probably because linen has the slightest abrasive function, while leather has almost none. Some people go berserk over such a statement, but prof. Verhoevens studies with aid of 3000X scanning electron maginification are pretty formal. Clean leather was not able to offer any significant reduction of solid burs present at an edge. I've had passionate discussion with a woodworker who claimed that Verhoeven must be a fool because he (the woodworker) used a leather bench strop to knock off the bur that developed during the sharpening process of knives and plane blades. Let the record show that ripping off a large and weak bur is not the same as abrading bur-like remnants that are much smaller and solidly attachted to the edge. Everyone who ever compared breaking a long stick in half with breaking a much shorter piece of the same stick, knows that breaking something off is not the same as abrading it.
Anyway, it seems that linen offers an advantage when it comes to edge maintenance work that requires, however slight and gentle, abrasion.
What linen doesn't do so well, is aligning the edge. I have shaved with a freshly honed edge, only stropped 120 laps on linen (instead of my regular 60 linen/60leather), and there was definitely something lacking. (this experiment can easily be copied). I think the structure of fabric is too rough to yield a perfect alligment of the very edge. (I actually prefer the term "fin" but it seems to confuse people).
Leather is the perfect material to allign the sub-microscopic ridges that are present on the very edge of a razor. It is not that good at cleaning the edge, nor can it be so easily cleaned as linen once it's contaminated.
That is my personal theory. I am the kind of person that needs a plausible explanation before I can agree to do something in a particular way. What I described above (and in the stropping article) works for me, and I like the results. I am sure a lot of fine people do something entirely different and like it that way. Shaving the artisan way is all about enjoying the process, not about doing something right or wrong. I can only testify about my way. If that helps some guys to overcome certain problems, that's just great. But it's certainly not my intention to offend other guys who are perfectly happy with doing something entirely different.