I looked pretty hard at linen a few years back, but only on walk-ins at stores and nothing online. I'm sure my impressions reflect only what the stores happen to be carrying and little about what I might learn if I did a whole bunch of ordering just so I could pinch and squeeze the goods. But here's what I came away with.
The linens available for fold and sniff in stores weren't all that thick, only going up to .013 on my mic as I recall. Seemed a lot tougher than cotton, so I picked up some .08 to try in a gun that needed a thin patch for the balls I had available. It was a patch blowing monster using cotton patches, but I didn't want to order a smaller mold if I didn't have to. Tried the linen and problem solved for about $6 as I recall, and I'm still using some of that original purchase years later.
Not seeing any heavier stock but curious, I doubled up the .08 and shot that in another rifle that liked .015. Bingo. Recovered patches were in lots better shape than ticking, and if I was a better shot I might have claimed better accuracy. But I wasn't about to go through the folferol of setting up a benchrest, so all shooting was offhand.
Couple of sips from a bottle doesn't tell you much, but that's what I got about linen. Sure seems like the answer any time a guy needs thin patches and cotton won't do the job. Looks like it's dandy for thicker patches too, but I got a problem there. Our local Chinamart shut down their fabric department and I was able to buy a whole bolt of cotton ticking for $1 a yard. Even using it for big projects like bag liners, I figure there's enough there for my great-great grandkids someday to be using, so there's just no sense buying more thick patching any time I can make the ticking work. Doesn't say a thing about how well thick linen will work for patches too. Just sezz I'm cheap.