Galileo Bad, Archimedes Good

Galileo's argument is that the theory where heavier objects fall faster is inconsistent a priori, because affixing a small stone to a larger stone would cause the composed object to fall faster than the smaller stone was falling when it was free. However, there is no logical contradiction here: what could happen is that the combined object would have an acceleration that is the (weighted) average of the acceleration of the components - slower than the lighter object but faster then the heavier object.

In fact, this is exactly what happens in an electric field: if you have two objects with the same mass but different negative charge moving towards a large positive charge, they will accelerate at different rates (the one with the bigger negative charge will "fall" faster). If you then tie the two objects together, you'll get a combined object that has more mass and more charge; the total electric force will increase, but its larger total mass will mean that it accelerates less. Alternatively, you can explain it as the less charged object dragging the heavier object down, such that the combined object moves at an average of their speeds.

The fact that this doesn't happen with gravity is a very special property of gravity, that only experiments can prove. A priori, gravitational mass/charge could have been entirely unrelated to intertial mass, just like electrical charge. Only much later, with Einstein's general relativity, did we get an explanation of gravity that makes this more than a coincidence - and it turns out that gravity is not a force at all, at least not one that acts on objects.

It still took like a thousand years for this to be experimentally demonstrated.

The important part of this "thought experiment" in the history of science is that it is part of the shift to empiricism that really drove science. It was important to go from "Well they were smart and they said, so it must be true" to "I don't care how smart you are, what you say doesn't match the data"

This is important, because "smart" people like Archimedes said a lot of stuff that was never true, but was taken as true for a millennia, often because it "sounded" right or obvious. More importantly, Archimedes could have done the exact same experiments that Galileo (and others) used to demonstrate he was not correct. There was no technological advancement required. He didn't, because the philosophy at the time was to "just think really hard about it" and "reason from first principles" and you would obviously get the right answer if only you are smart enough. Who needs data? You're smart and you thought hard about it, so you cannot be wrong!

People should recognize how important that is to remember in the current world.

Pre-university schools, even today, focus on teaching practical math. Most people can get by just fine without skills in abstract math, theorizing, and proofs (though those skill would make a lot of people much better at whatever they do).