If other people have this issue, I would try supergluing a screen protector over the old screen - leave a wide margin around the edges, home key, ear speaker. You could even just cut a piece of the protector into a square, or get a rectangle of shipping tape, the glued down portion needn't cover the whole screen. Then, glue the suction cup on top of the glued down screen protector or tape. I suggest this over the utility knife because it seems like the knife method would at the very least scratch up and nick the frame.

I suspect that the age of the phone and accident that caused cracks make this method possible for some people. My mom's phone was only a month or so old and the suction cup only caused the screen to raise slightly. There was certainly no crack I could squeeze a spudger into. I just superglued the suction cup onto the screen in the end, which was very effective.