One thing some people often repeat is that PoS coins won’t last, because they lose the network security a vast number of miners provides. Can you explain how the network will remain secure and thriving?

This is not true. Similar to PoW that’s subject to a 51% attack, the PoS will need to hold 51% coins in order to be able to attack the coin. But holding 51% and then attacking your own coin normally is not in your interests.

Do you anticipate people to keep mining once block rewards drop to 1? Why keep block rewards for mining going, if no one will point machines toward it given the low reward?

If no one will mine blocks at 1 coin per block, we will happily mine them. Though not much, if no one competing with you, you still can get a lot.
Even no one mines, it doesn’t matter. As the transactions will be ensured and confirmed by the PoS block generations. The coin network will still go on fine, and remember that we will run a few servers to ensure a minimal service at any time.