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

tools/ci_build/github/azure-pipelines/npm-packaging-pipeline.yml

@@ -44,19 +44,17 @@ stages:
     PoolName: 'Win-CPU-2021'
     PackageName: 'onnxruntime-web'
 
-- stage: Build_React_Native
-  dependsOn: Extract_commit
-  jobs:
-  - template: templates/react-native-ci.yml
-    parameters:
-      NpmPackagingMode: ${{ variables.NpmPackagingMode }}
-      BuildConfig: 'Release'
-      PoolName: 'Linux-CPU'
-      PackageName: 'onnxruntime-react-native'
+- template: templates/react-native-ci.yml
+  parameters:
+    NpmPackagingMode: ${{ variables.NpmPackagingMode }}
+    BuildConfig: 'Release'
+    PoolName: 'Linux-CPU'
+    PackageName: 'onnxruntime-react-native'
+    BuildAndroidAARStageDependsOn: 'Extract_commit'
 
 - stage: Download_Node_Package
   dependsOn:
-  - Build_React_Native
+  - ReactNative_CI
   - Build_web_Release
   - Build_web_Debug
   jobs:

tools/ci_build/github/azure-pipelines/templates/react-native-ci.yml

@@ -19,10 +19,15 @@ parameters:
   displayName: 'Package name'
   type: string
   default: 'NPM_packages'
+- name: BuildAndroidAARStageDependsOn
+  displayName: 'Stage the Build Android AAR stage depends on'
+  type: string
+  default: ''
 
 stages:
 - stage: Build_Android_AAR
   displayName: Build Android AAR
+  dependsOn: '${{parameters.BuildAndroidAARStageDependsOn}}'
   jobs:
   - template: android-java-api-aar.yml
     parameters: