Here is the happy conclusion to this saga…
After running speedtest-cli and iperf3 network throughput tests which showed abysmally slow speeds between the droplet and the outside world, I asked DigitalOcean to investigate and they concluded after doing their own testing:
We have discovered some issues with the hypervisor where your Droplet is located. We are working with our backend team to migrate your Droplet to another hypervisor.
All is well again.