[7.17](backport #30795) ci: refactor pushDockerImages main pipeline

[mergify]

This is an automatic backport of pull request #30795 done by Mergify.
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[v1v]

#28286 was not backported to 7.17, so changes are a subset from the original PR

[elasticmachine]

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• Start Time: 2022-03-17T08:58:17.969+0000

• Duration: 76 min 42 sec

Test stats 🧪

Test	Results
Failed	0
Passed	34169
Skipped	3928
Total	38097

💚 Flaky test report

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[v1v]

/test