I discussed this very topic with a couple of other freepers just a few days ago. Their position was the same as yours. Is there some website out there with your account of things? Anyway, here's my view:
Galileo's evidence was just fine. A prediction of the Copernicus theory was that Venus would be seen to go through phases. This was visible with Galileo's telescope.
The visible evidence of the phases of Venus was an extremely powerful confirmation of part of the Copernicus model. That evidence made it undeniable that Venus orbited the sun. But that didn't show anything about the Earth's movement. Specifically, the phases of Venus didn't rule out hybrid models that allowed for the planets to orbit the sun, while the sun and everything else still orbited the earth. There were such models at the time.
Galileo's observations, however, took away yet another argument for a stationary earth -- one which has been largely forgotten. It had been argued that the earth must be fixed in place because if it moved, it would leave the moon behind! That sounds goofy now, but Galileo flourished a generation before Isaac Newton, and in Galileo's day, no one realized that gravity held the moon in its orbit around the earth.
What Galileo argued here was a deduction that followed from his discovery that Jupiter had moons. Those moons clearly orbited Jupiter, and somehow they didn't get left behind, even though it was obvious to all that Jupiter was moving. Thus, although then inexplicable, our moon's similar behavior couldn't be advanced as "proof" that the earth was stationary. Therefore, Galileo's work left no argument remaining that the earth was stationary -- except the then-current interpretation of scripture.
This was more than enough evidence to be persuasive. Isaac Newton didn't add anything specific to this, at least not that I'm aware of.