Here’s a simple model of the slippery slope. You have to adopt a position on an issue and defend your position to yourself and your critics. The spectrum of positions ranges from the left-most extreme to the right-most extreme and you have to decide whether to take one of these extreme positions or some moderate point in the interior.
Defending a moderate position is a delicate balancing act. It’s a very special set of utility functions which attain their maximum right at that point, and you need to convince your critics that the right utility function happens to be one of those. Any slight perturbation of a utility function in that set will push you to the left or right so your critics have an easy task. And once you’ve lost the first battle your credibility is damaged.
The easiest positions to defend are the extreme ones. At an extreme position you have a binding constraint. To defend your extremist position it is enough to say that you are such an extremist that you would like to move even farther to the right if that were possible. The set of utility functions that have an optimum somewhere to the right of the right boundary is a large set. You can perturb such a utility function and the extremist position will still be optimal.
The same logic explains why a few special interior positions can be robust to the slippery slope. Think of a kinked budget constraint. A large set of utility functions achieve their optimum at a kink.