Fix Npm packaging pipeline (#15425)
### Description
It seems like https://github.com/microsoft/onnxruntime/pull/15329
re-worked some jobs in `react-native-ci.yml` into stages. When this
template is used from within `npm-packaging-pipeline.yml`, there is
problem in that there is a stage that contains multiple stages as jobs.
Per my understanding, this is not acceptable to Azure DevOps. So,
re-working some portion of `npm-packaging-pipeline.yml` to accomadate
changes in https://github.com/microsoft/onnxruntime/pull/15329
### Motivation and Context
Fix NPM packaging pipeline
Validating test run with fix:
https://aiinfra.visualstudio.com/Lotus/_build/results?buildId=297391&view=results
