I feel that the word happiness in itself by definition confuses the concept in a way that it has become ego-centric. I also feel that the idealized relief from suffering in its relation to physical or emotional discomfort (is there truly a difference?) is misguided.
I think of a three legged dog - from my perception, I’ve never come across a three legged dog that seemed to care, acknowledge or consider that their three legs left something to be desired. Surely they are at some level aware that they have only three legs as they have adapted to life with three legs, but it seems (or I’d like very much to believe) that they do not have three legs or four or eight, or that their number of limbs differ from other dogs, they just have legs that propel them along the path they choose.
To the contrary, whenever I have a broken or damaged body part that invokes a limit, I desire a reality that does not exist where in that moment I do not have said injury because I feel that is the cause of my suffering, when in truth the desire itself is the cause. This is my attempt at explaining my conceptualisation of that which equanimity and happiness to me seem insufficient.

The goal to ‘disidentify’ with ones emotional or physical experiences as @Coben touched on seems to be a flawed goal on both a human and conscious level. The flaw on the human level has been addressed so I’ll focus on why I believe it to be so on a conscious level. On a conscious level, I believe most schools of thought agree that experience is imperative in the role of consciousness. To attempt to minimize or dampen the experience therefore, seems contrary the the concept or purpose of consciousness. I feel the more realistic path to attaining the concept which seems to be the topic of this thread is to fully accept our experiences without desire for what is not, and to fully replace judgment of our experiences with understanding.