And doesn't that spring on the opposite side not only hold the slide stop in place in the frame but also move it to the ready to use (down position so it drops out of the notch in the slide.)
Is that spring still there/in the right place?
A CZ slide stop (not sure what model). See the groove in the top just back from the end of the shaft? That's the groove that spring leg sits in. Holds the slide stop in the frame and keeps it "down" so it only catches the slide notch on the left side of the slide when the follower of an empty magazine pushes the slide stop up
Here's a picture of the inside of a CZ75 slide (CZ75 Compact converted to use CZ85 slide stop/release levers). You can see that small diameter (but pretty stout) spring on the right side of the frame where it presses against the slide stop shaft.
If that spring isn't there, or has broken, become weak for some reason, then it could be causing your problems. It that spring is "right" it can't be really difficult to press that slide stop over to the left far enough to get it out. I use the bottom corner of a plastic magazine base on mine to get that sldie stop shaft to move far enough for the spring top pop up out of the groove. I cannot do it with a finger tip or my thumb.