I recently witnessed an attempt to bring a larger hard material crate on a flight by purchasing a seat for it. I was seated in a row 1 aisle seat, so saw, but didn't always hear the interaction. When the passenger got to the plane door, the flight attendants seemed unsure what was allowed, and while they seemed to be consulting someone by tablet and phone, they allowed the passenger to board with the crated dog. The crate was large enough that it had to be carried down the aisle vertically.
After boarding was complete, the flight attendants must have gotten official guidance, and the passenger and dog were deboarded. The crate looked like it was too big to fit belted into an economy seat, even vertically. Take-off was delayed while baggage was unloaded, but with a significant tailwind we still arrived at our destination 45 minutes early.