Triggering a build not only on pushes to a branch, but also on a deployment would be great.

Webhooks exist and competing CI services support them, cf. e.g. the https://deploybot.app/ GitHub App to manage deployments. It advertises support for different CI services.

GitHub API docs:

* https://docs.github.com/en/rest/deployments

* https://docs.github.com/en/developers/webhooks-and-events/webhooks/webhook-events-and-payloads#deployment