But a more complete notion of "object" isn't what solved the problem.
Much of what constituted physics prior to the advances by Galileo and Newton (and many others) was essentially what was developed by Aristotle. Within the Aristotelian framework, which is what most educated people knew at the time, the concept of a distinct "object" is perfectly well defined. My point is that the natural philosophers at the time were quite capable of asking the question, "do heavier objects fall faster?" without being led astray by an incomplete notion of "object." In other words, the problem is conceptual, not semantic. The Aristotelian - again, the dominant framework back then - was instead led astray by the notion that all objects had a "natural" tendency to move towards the center of the Earth, and that this tendency was stronger for heavier objects. This worldview had a very long history, and a fair amount of evidence (such as it was) to back it up.
In this context, what matters is not the distinct objects but instead what properties influence their dynamics. It should be clear then why it was necessary to demonstrate the empirical truth. Many plausible theories had emerged, including the prevailing Aristotelian one, which accounted for the differences in, say, a lead weight falling versus a feather. Or, to get back to your original example, the Aristotelian would have answered, probably confidently, that yes, your chain of cannonballs would fall faster than separate ones. Your (correct Newtonian) intuition is that the mere act of chaining them together should not alter them, given that the threads would not exert forces in free fall, and therefore mere mass should not alter the falling rate. But, those notions depend entirely on the Newtonian framework. "Force" and "free fall" are the undefined concepts, not "object."
A key insight was needed, namely that you could separate out different causes in the motion; in this case, gravity and air resistance. (Though, of course, "cause" is itself a thorny concept...) Galileo's experiments demonstrated the point elegantly. More formally, this resulted in the development of the notions of inertial reference frames and linearity, which underpin much of classical physics.
I hope you don't think I'm belaboring the point. Newton's revolution was so total, so complete, that it changed everything about how we think about physical problems. Centuries of difficult philosophical and scientific work has been condensed down into a semester of freshman physics. To us, the problem is trivial, and it is easy to think that such experiments are so trivial as to be useless. To Galileo, this would not have been the case. A single thought experiment would not have been enough back then, even though it suffices now.