That's a good summary. I would only add that Brahe himself developed a sort of "compromise" model between the Ptolemaic and Copernican systems in which the planets (aside from the Earth and the Moon) all revolved around the Sun, which then revolved around the Earth. The Tychonic system explained the lack of observed parallax, but kept the computational simplifications of the Copernican system. This was the model preferred by most of the learned men in the Church at the time.