What might the miners do at this junction?
Sure, they still get transaction fees. Would those escalate?
Assuming the block size remains limited, fees should escalate over the same time that the subsidy drops. Hopefully, fees rise to an extent that miners continue to secure the chain without drastic drops in hash rate.
If the economic design looks to be failing, I'm sure we'll see it reflected in the hash rate long before the last coin is mined.
Or a reason to change the reward algorithm?
That's always a possibility, but I'm hoping the fee market created by limited block size will prevent the need for that.