when I start a client in the stabilization phase, I usually also include some strength exercises that are supported to keep the client engaged. As I observe better control, I add some instability to the stabilization exercises and then watch again to see whether those have been mastered as well. I then phase out some of the stabilization and add more strength endurance but I never really leave the stabilization phase entirely and always challenge my clients with exercises in the category.
To note that I rarely go beyond stabilization and strength endurance because of the kind of clients I have. When a client comes back after an illness or injury. I step right back into that phase again in which I also include MELT and many exercises from the BioMechanics Method.
Hope this helps.