The closest you will find to Galileo in ancient times is one of Aristotle's successors, named Strato of Lamsacus. The Wikipedia article on him explains that he discovered that falling bodies accelerate as they fall, and one of his most convincing arguments for this is because water in a column falling down as a stream breaks into droplets after a certain distance, and this is clearly due to the increasing speed making the column of water thinner and thinner in a regular way.
Since he doesn't seem to have investigated the law of acceleration in any quantitative mathematical way, it is difficult for him to notice a mathematical regularity, even one as obvious as all objects picking up speed at the same rate in time. Galileo probably noticed the universal acceleration empirically before coming up with the thought experiment you describe, which suggests that this universality is exact.
Just because an argument is obvious in hindsight doesn't mean that people think of it. There are countless simple things the ancients could have done, but didn't. For example, Copernicus made an argument that the stars must be very far away in Earth radii, because the time of star-set is exactly 12 hours after star-rise (this is not quite right because you need to correct for atmospheric refraction, but this is also an effect that could have been measured in ancient times, if they had a pendulum clock, something they also could have, but didn't, construct. They had the required differential gearing in Archimedes' day, see Antikythera mechanism, but forgot about it later).
The ancients could have measured the positions of the planets as accurately as Brahe, but didn't. Other things that the ancients missed:
- No slip boundary conditions for fluids are required by continuum dynamics, and are violated when you have a contact line between three bulk materials, like when a drop sticks to metal in air. The air is completely displaced where the drop hits, and this is evidence for atomism.
- There is an upper limit to the degree to which oil spreads on water (Benjamin Franklin's early estimate of Avogadro's number). This is a convincing demonstration of atomism.
- Parabolic mirror telescopes (discovered by Newton, available to Archimedes, since he knew the focusing properties of parabolic mirrors) One can make a perfect parabolic reflector by spinning molten metal as it cools.
It is not easy to say why people miss things. The atrocious politics of the ancient world, which put people in hereditary heirarchies of power and made slaves of the majority, might be the reason scientific discourse died, because nobilities tend to make a nobility discourse and a nobility philosophy to justify their status, and science can only proceed with slave language and common-sense slave arguments.