The ferrets dying was many years ago, when they were trying to make a vaccine for SARS (a closely related coronavirus to COVID).
They used a whole virus approach for that failed candidate.
The whole virus approach was abandoned for coronavirus vaccines over a decade ago, because it was found to be (dramatically) unsafe.
Researchers have sought to find just the minimal features of the virus (epitopes) that would stimulate effective immune response, while minimizing undesired side effects. They nip those features off, and graft them onto innocuous common viruses as carriers (in the case of J&J, AstraZeneca, Sputnik V, etc.).
A newer approach, is to use an mRNA blueprint of just that tiny snippet of the locking mechanism at the tip of the spike protein (epitope), to induce cells to manufacture perfect fresh copies on site in the body. That is what Pfizer and Moderna do. There is absolutely no other viral material to cause potentially unwanted side effects (for safety), and you can get a higher number of the epitopes of interest for the immune system to recognize (for effectiveness).
The epitopes produced, and the mRNA instructions to produce it, have a relatively short existence in the body (about a ten hour half life for the mRNA). Then they degrade and are disposed of, like any other dead cells or extraneous material. Neither the small snippet off of the COVID spike protein, nor the mRNA instruction set, have any reproductive ability.