From my (ignorant, despite being interested in prions for decades now) perspective, you have something that (1) can replicate, (2) is durable (3) is harmful, and only the relative rarity of prions in general (because 1 and 3 are physically unusual to combine) have kept them from being a general problem. As we tinker with DNA and thus look into creating new proteins and factors that would influence protein-folding, I'd think it's worth taking the matter seriously (though not fear-mongering), certainly as part of our need to understand proteins better anyway.
What are my faulty assumptions?
A virus can overtake a cell and mobilize its resources to create a bajillion copies of that virus. A prion can transform only as much copies of that protein as that cell has made in its normal processes.
A virus requires a cell to replicate, bacteria require a much more limited range of hospitable environments.
Both bacteria and viruses "attempt" to create exact copies, just as a prion does. Admittedly prion replication probably has more accuracy, but that prions occur at all indicates the process allows for natural selection.
These don't seem to put prions outside of the "replication" category - they replicate by encountering other proteins, so they are in a body. Their rate of replication is limited (or boosted) by the rate of encountering these proteins. They attempt to create copies of themselves with the chance of failure or different end result.
They can't reproduce outside of the body, but neither can viruses.
Not really. The point your parent is making is that prions themselves don't reproduce like viruses and bacteria. Its basically impossible for a large scale breakout to occur and spread.