Createasaurus, I agree, that solution could get messy quickly. I wish there was a box somewhere that I'm just forgetting to enable, but it seems that's not the case.
Given that it's not an ideal solution, I did give what you said a try, and it does work. It's not
that messy, but it does feel a little bit iron-fisted. Then again, my game is already filled with iron-fisted solutions!
Basically, my moving platform has an empty gameobject attached. This gameobject has an "is trigger" collider that is the width/depth of the platform, and much taller.
When an object tagged "player" triggers the collider, it passes to a state where the camera object and the character object (both containing within them smaller objects, FSMs, etc) are both parented to the platform. This was necessary because my character relies on the camera for movement relative between the two. This also causes the camera to rotate with the platform, which is slightly disorienting, which was desired, so yay for two birds with one stone.
When the collider senses an object tagged "player" exiting the collider, it passes to the original state which sets the parents of my camera and character objects to "none".
I'm also fond of this solution because I can use a similar FSM for my moveable puzzle cubes, and this enables me to forget about rigidbodies on my platforms and character, which was causing a whole slew of other issues. My second post, with that error message, went away when I removed the rigidbody component from my moving platform, by the way (in case anyone searching for that is brought here).
These might change based on your camera/character setup, but I went through a few iterations that I imagine would each work for different setups. Let me know if you want to try this solution (despite its slight complications) and I can help you try to work through it if you hit any snags!