In simple terms, "Core Stability" is like your lives counter. It depletes when you lose all "Integrity", and some theme-related effects trigger based on how low it is. However, if you're doing very well and avoid dying (losing all "integrity"), you never get to interact with this mechanic. Seasoned shmup players are understandably confused. It is a problem I would like to find a solution for, but for less experienced players that definitely die at least a few times - the mechanic works as intended, so I think it serves its purpose for the scope of the jam.
Hard mode has been designed with Yellow in mind, which is why the switch after Stage 2 occurs, it was just not communicated very well. In the future, I would like add separate Hard mode stages for each character. Thank you for playing, I'm glad you enjoyed the game.