I've heard that if the cat learns to flush, it will flush your toilet all day because it's fun.

Kids do this and cats are smarter.

Some cats learn to flush, which is absolutely nuts, but it happens.

apparently the cat can learn to flush if the cat hits it by accident

My cat learned how to turn on the tap nearly wrecking the well. Now he's an outdoor cat. Seems happier too.

You could technically automate it. Training the cat is the hard part. Detecting that a cat (or frankly, anything else) is on the toilet is easy with some electronics.