I may be the one person thinking this, but I do notice a difference in horse butt vs. a softer high draw cow leather.

But it's not on the first shave.

It's on how long the strop will keep the razor in shape. The harder the strop, the longer I've had luck keeping the razor off the hones. The longer the razor stays off the hones, the longer the razor lasts.

I have a kanayama in the closet, but haven't used it yet because it's supposed to be for Christmas. It took a while for my horse butt strip strop to break in and feel smooth and compliant on the surface, it'll be interesting to see how the kanayama compares. I haven't used any of the high $$ harder cowhide strops, either, I'd probably find them favorable if they were harder with less draw, but horse butt strips are so cheap that I can't justify buying anything other than cordovan if I'm going to spend money.

If I were tight on money, I would personally just stick with plain untreated and undyed horse butt strips, cut to a hanging strop size. The cost per strop is about $10 plus hardware for a $2x24 strop if you get decent horse butt.

Any clean strop though, soft or hard, should do a good job of improving an edge right off the hones, though. I am by no means saying that anyone should buy a kayanama strop or a horween shell thinking they'll get a sharper razor right off the stones, and it's possible I might find those expensive strops less favorable than a cheap and already broken piece of horse butt.