[Remote Routing Table] Add write flow for remote routing table

[himshikha]

Description

This change adds write flow for uploading routing table to remote store during cluster state publication.

This is dependent on base PRs #13304 and #13577

Related Issues

Resolves ##14033

Check List

• New functionality includes testing.
  • All tests pass
• New functionality has been documented.
  • New functionality has javadoc added
• API changes companion pull request created.
• Failing checks are inspected and point to the corresponding known issue(s) (See: Troubleshooting Failing Builds)
• Commits are signed per the DCO using --signoff
• Commit changes are listed out in CHANGELOG.md file (See: Changelog)
• Public documentation issue/PR created

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[opensearch-trigger-bot]

The backport to 2.x failed:

The process '/usr/bin/git' failed with exit code 128

To backport manually, run these commands in your terminal:

# Navigate to the root of your repository
cd $(git rev-parse --show-toplevel)
# Fetch latest updates from GitHub
git fetch
# Create a new working tree
git worktree add ../.worktrees/OpenSearch/backport-2.x 2.x
# Navigate to the new working tree
pushd ../.worktrees/OpenSearch/backport-2.x
# Create a new branch
git switch --create backport/backport-13870-to-2.x
# Cherry-pick the merged commit of this pull request and resolve the conflicts
git cherry-pick -x --mainline 1 1084ba9b167af8342be0acea075c83fa1673dcb1
# Push it to GitHub
git push --set-upstream origin backport/backport-13870-to-2.x
# Go back to the original working tree
popd
# Delete the working tree
git worktree remove ../.worktrees/OpenSearch/backport-2.x

Then, create a pull request where the base branch is 2.x and the compare/head branch is backport/backport-13870-to-2.x.