Your description of what happened during inflation is basically correct. However, in order to explain why a black hole didn't form as soon as inflation ended and gravity started behaving "as we know it", you have to include the fact that all of the matter that was formed at the end of inflation was expanding very, very rapidly. (The universe was actually vacuum during inflation--at the end of inflation all of the energy in the false vacuum got transferred to the Standard Model fields, creating matter and radiation at extremely high temperature and expanding.)
