The middle position that works, and what makes the rat argument real or not:
Leave sound fruit, but not in a heap, and not against the trunk. Scattered fruit that is accessible to birds and dries or breaks down quickly is not a rat problem. A pile is — it is food plus cover, which is the combination that matters. Rats want somewhere to be while they eat.
Keep the base of the trunk clear regardless. Fruit against bark holds moisture on it all winter and that is its own problem, separate from disease and rodents.
Take the worst of it away in autumn, then stop worrying. You do not need to clear every apple. You need to remove the diseased ones and avoid creating a heap.
If you have more sound fruit than the ground can absorb, the other honest option is to pick it before it falls and give it away. A tree that drops that much is usually producing more than the ground under it can process.