Over the years, I picked up 4 or 5 of the “screwless’ vises, since then I sold off 2-3 of them, because, I find them a PITA to use. I’ve owned a Vertex and so others and they all seem the same. Everything I put in the vise, seems to be s I’ve where, its in between the notches for the clamp mechanism. I used it for a while, with a piece of coat hanger, to hold the clamp mechanism in the proper position, a lot of fiddling around. It became easier to get a regular screw type vise.
As to the OP’s original observation, all of my drill press vises are made the same way, with a groove at the bottom fixed jaw, That make thin parallels unusable, so I always used 1/2” thick parallels.