Maybe you could figure out a way to only add that effect if you're too close to the back edge of the platform as it moves, and add the screen shake to accentuate the effect occurring.
E.g. if the platform is moving left and the player jumps onto the right side within say, 48 pixels of the right edge, the player character would stay stationary and fall off the edge if the player doesn't move while the platform moves and vice versa.
If the player is between the 48 pixel limit and the other edge, then the player moves with the platform.
The player, would then need to move to the centre of the platform, or risk falling off the edge after jumping onto it.
In reality, for when you jump onto a moving object travelling at a fixed speed, it is only the initial acceleration to the speed of the object you're on when you would find it difficult to balance, and likely fall off.