No experimentation required
"Galileo's suspicion was that all objects (in the same gravitational field) should fall at the same speed, irrespective of their mass, and he was eventually able to conclude that in a vacuum this would indeed be indeed be the case."
It wasn't really a suspicion that required observation to confirm or confute. It was based on reasoning alone, based on what would happen if a lighter (supposedly slower-falling) object were connected to a heavier (supposedly faster-falling) object.
The lighter object would retard the fall of the heavier object, so the speed of the pair would be a bit slower than that of the heavier. Yet by the same principle, the two connected objects would now constitute a single heavier object, so would be faster than both.
It's clearly impossible for both conclusions to be true, so the principle ("heavier objects fall faster") that leads us to them can't possibly be true.