Who was doing very good physics work before Galileo?

Lots of people! Oresme, Al-Brituji, Philoponus, Copernicus, Tycho, Abul-Barakat, Buridan, Grosseteste, Bacon, Al-Haytham, and Oughtred are all examples.

The basic laws of mechanics were unknown before the 17th Century, and the Aristotelian physics people believed in was completely wrong on the most basic physical principles.

This is a common misconception. There's this narrative where physics remained completely at a stand-still between Aristotle and Galileo. But a lot was being done. Most prominently, the theory of impetus was developed, a precursor to the more modern notion of momentum (as laid out by Galileo and some of his contemporaries). Without the theory of impetus as a guide, Galileo would never had made his discoveries.

Galileo did two things to initiate the modern study of physics. He made the first telescopic observations, finding strong evidence for heliocentrism, and realizing that there is some sort of influence of the Earth on the Moon, Jupiter on its moons, and the Sun on the planets.

False. Thomas Harriot made telescopic observations at least four months before Galileo. Moreover, the idea that local bodies would influence their neighbors. Oresme for example raised this possibility 3 centuries before Galileo.

He also conducted experiments into falling bodies on Earth, formulating a rudimentary version of the law of inertia

Which as an improvement on the pre-existing theory of impetus.

There's no question that Galileo was an absolutely top-notch physicist. But the idea that he initiated modern physics ignores both home much work was done before him and how much other work was done contemporaneously as he was. You may want to read more on this subject. One place to start would be Freely's "Before Galileo: The Advancement of Science in the Middle Ages" which I haven't read but seems to have very good reviews.