I use a Selway type stringer for all my bows. I have zero problems with the rubber block sliding on either the longbow or recurve. The trick for me is to leave the string full length and spread both feet as wide as possible so the string is pulling nearly straight down on the limb.
What scared the living daylights out of me was borrowing a longbow stringer for my recurve when I ran off without my stringer. That TINY little cup was holding the tip of my upper limb by a hair!
I'd still rather do it that way than step through the bow, but I had the feeling that little cup could slip off my bow tip at any time. That would have been bad!
With the cup and block stringer, the worst that can happen is that the block slides back down the limb as I lift the bow and that is a gradual thing - not a sudden "there she blows!" experience. The only time I had the block slip was when I shortened the string so I could stand on it with only one foot and that puts the force at an angle back toward the riser. With the full-length string and 2 feet spread apart, the block pulls the limb tip straight down - no slip, no twist- works great for me.