[AIP-94] Migrate airflow providers command to the API server

[henry3260]

What

• Migrated the legacy airflow providers list and airflow providers get command to retrieve installed providers through the airflowctl API client.

related: #68402

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Was generative AI tooling used to co-author this PR?

• Yes (please specify the tool below)

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• Read the Pull Request Guidelines for more information. Note: commit author/co-author name and email in commits become permanently public when merged.
• For fundamental code changes, an Airflow Improvement Proposal (AIP) is needed.
• When adding dependency, check compliance with the ASF 3rd Party License Policy.
• For significant user-facing changes create newsfragment: {pr_number}.significant.rst, in airflow-core/newsfragments. You can add this file in a follow-up commit after the PR is created so you know the PR number.

[github-actions]

Hi maintainer, this PR was merged without a milestone set.
We've automatically set the milestone to Airflow 3.3.1 based on: backport label targeting v3-3-test
If this milestone is not correct, please update it to the appropriate milestone.

This comment was generated by Milestone Tag Assistant.

[github-actions]

Backport failed to create: v3-3-test. View the failure log Run details

Note: As of Merging PRs targeted for Airflow 3.X
the committer who merges the PR is responsible for backporting the PRs that are bug fixes (generally speaking) to the maintenance branches.

In matter of doubt please ask in #release-management Slack channel.

Status	Branch	Result
❌	v3-3-test

You can attempt to backport this manually by running:

cherry_picker 17aa541 v3-3-test

This should apply the commit to the v3-3-test branch and leave the commit in conflict state marking
the files that need manual conflict resolution.

After you have resolved the conflicts, you can continue the backport process by running:

cherry_picker --continue

If you don't have cherry-picker installed, see the installation guide.