Modern tools should be able to compensate for much of the lack of genetic diversity.
Minimum genetic viability is something of a myth.
Bottlenecks basically means an increased degree of incest (no not direct incest but direct incest is the most extreme form).
This exposes recessive traits both positive and negative. In the cruel light of natural selection recessive flaws are eradicated.
The population itself doesn't die, it is purified in a sense.
Or in other words if one man and one woman are the last two left in North America, their kids will have to reproduce with each other. It's not genetically possible they are incapable of producing children without a genetic flaw (or else they would express the flaw themselves).
So worst case 1/4 of their children (on average) have the negative trait. If it was some central European dynasty where that one kid with the recessive flaws HAD to be heir things might get worse. In the real world though if those children are male they'll die or father no children. If they're female they'll die and mother less children.
This isn't just genetic theory, many small islands have been populated from seed populations of only only a handful of humans and instances where non-human animals populated the island have been seen where the original population was only two.
I picked 500,000 because I can't fathom a disaster that wouldn't leave that many alive (so long as there was still a biosphere). That's probably a good estimate for the number of people in complete isolation who wouldn't be touched by the most deadly and virulent disease as well.
People who still believe climate change of a few degrees or elevated radiation from either the sun or nuclear weapons could possibly cause that many deaths have zero concept of science, geology, and earth history.
Agreed.