Fleet-Server Elastic Agent can be unhealthy for a few minutes after enrollment.

When starting the Elastic Agent running Fleet Server and Elasticsearch simultaneously, like [`elastic-package`](https://github.com/elastic/elastic-package) does, the Elastic Agent running Fleet Server can be unhealthy for a few minutes right after the whole stack goes up.
The `elastic-agent status` command returns healthy while Fleet UI still shows unhealth.

Video showing the issue:

[fleet-server.mp4](https://github.com/user-attachments/assets/3cbf6855-9b28-4d9f-8990-dc0b7ad0b04e)

Agent JSON from when the Elastic Agent is showing unhealthy: [agent-JSON.json](https://github.com/user-attachments/files/26314749/agent-JSON.json)

Diagnostics: [elastic-agent-diagnostics-2026-03-27T13-51-25Z-00.zip](https://github.com/user-attachments/files/26314748/elastic-agent-diagnostics-2026-03-27T13-51-25Z-00.zip)

Start by reproducing the startup sequence used by elastic-package with Fleet Server and Elasticsearch running simultaneously. Compare the `elastic-agent status` result with the Fleet UI state, then inspect the attached agent-JSON.json and diagnostics archive for the health transition. Done means the Fleet UI reports the expected health promptly after enrollment, consistently with the command output.

Written by the indexing model from the issue text.