[8.13](backport #39239) Update x-pack/heartbeat pipeline to match Jenkins

[mergify]

Proposed commit message

In scope of https://github.com/elastic/ingest-dev/issues/3254 - updated x-pack/heartbeat pipeline to match same steps & commands as in Jenkins.
Jenkins vs. Buildkite comparison: https://gist.github.com/oakrizan/ad3b982cd59e42d25d80f664a194998b
BK builds: https://buildkite.com/elastic/beats-xpack-heartbeat/builds?branch=oakrizan%3Axpack-heartbeat-fixes

Checklist

• My code follows the style guidelines of this project
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• I have made corresponding change to the default configuration files
• I have added tests that prove my fix is effective or that my feature works
• I have added an entry in CHANGELOG.next.asciidoc or CHANGELOG-developer.next.asciidoc.

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This is an automatic backport of pull request #39239 done by [Mergify](https://mergify.com).

[elasticmachine]

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[dliappis]

LGTM