No. Hydrogen ions , deuterium ions, helium ions, and traces of lithium ions were created in the first four minutes after the Big Bang. It took about 500,000 years for the universe to expand and cool to the point where the high photon flux would not reionize these ions. Thus at about half a million years, these atoms captured electrons and created the cosmic microwave background. The rest of your post is accurate. It is probable that we are third generation or further supernova remnants. The high levels of carbon, oxygen, silicon, and iron are easily accounted for in nuclear physics depending upon the sizes of the stars. Higher elements are only accounted for in supernovas because it takes endothermic nuclear reactions to make them. This is also why they are so rare, especially for high atomic numbers.
"Condense" was probably a poor choice or word on my part but I couldn't come up with better and it at least communicates the change in state.