The post receive hook provides you real-time feedback as it runs in the terminal where you did the git push. If something broke during the deployment you'd get notified by looking at the output. If it's running in CI, you'd see a CI failure and get notified using whatever existing mechanisms you have in place to get notified of deployment pipeline failures.

Zero downtime server upgrades are easy. You could make a new server, ensure it's working in private and then adjust DNS or your floating IP address to point to the new server when you're happy. I've done this pattern hundreds of times over the years for doing system upgrades without interruption and safely. The only requirement is your servers are stateless but that's a good pattern in general.