The barbers of old were probably working on simple carbon steels. The newer alloy steels, even O1, do benefit from having abrasive on the strop since they themselves are more abrasion resistant than simple carbon steels.
I have an old barber strop that I cut down for bench use. It is old and like nothing I have seen anywhere else, 4" wide, about 28" long and just over 1/4" thick, and only slightly less stiff and hard than a length of hardboard. No give in the surface and stiff enough in itself that it could have stropped at a pretty shallow angle when hung. The flexible strops I seen now are as flexible as a leather belt and in order to keep the strop angle shallow very little pressure can be applied. Some say you don't need pressure to strop, maybe, but Longstrider who is known for his strops, and his sharpening, swears by high pressure and his skills took him to the Bladesport cutting championship at the Blade Show a few years ago. Hard to argue with the performance of his edges.
For my sharpening a paddle strop is hands down better for what we do.
My paddle strops tend to have pretty thin leather, maybe 2mm, just enough for a little give and something to hold honing compound. They won't polish a large bevel in one pass at one angle, that is the effect of the thin leather, it results in a smaller contact area, but I think it gives me more control over the stropping angle, not just the angle that the blade is held at, but the angle taken by the strop material as it springs back as the blade passes over it. Very soft material springs back at a steeper angle for the same applied force. For convex edges the difference is minor and probably not worth worrying about, go with thicker and/or softer leather. For stropping a flat grind, like a scandi, or a chisel, I want as little rounding effect as possible.
