[DSIP-103][AlertServer] Embedding the AlertServer into the API Server

### Search before asking

- [x] I had searched in the [DSIP](https://github.com/apache/dolphinscheduler/issues/14102) and found no similar DSIP.

### Motivation

Currently, DolphinScheduler requires a separate alert-server to handle workflow and task alerts. Although the alert-server is lightweight, maintaining and deploying it separately adds operational complexity.

We aim to remove the standalone alert-server and embed its alerting functionality directly into the API server.

https://issues.apache.org/jira/browse/GSOC-306

### Design Detail

_No response_

### Compatibility, Deprecation, and Migration Plan

_No response_

### Test Plan

_No response_

### Code of Conduct

- [x] I agree to follow this project's [Code of Conduct](https://www.apache.org/foundation/policies/conduct)

The issue proposes embedding the standalone alert-server into the API server but names no files, entry points, design details, or tests. Start by locating the API server and alert-server components, then clarify the integration design, migration plan, and test coverage needed before implementation. Done should mean alert handling works through the API server without a separately deployed alert-server.

Written by the indexing model from the issue text.