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Clean up Azure clients

**What this PR does / why we need it**:

Follow up of #63278 (Use new Azure SDK APIs for load balancer and public IP operations), clean up all other clients.

**Which issue(s) this PR fixes** *(optional, in `fixes #<issue number>(, fixes #<issue_number>, ...)` format, will close the issue(s) when PR gets merged)*:
Follow up of #63278

**Special notes for your reviewer**:

**Release note**:

```release-note
NONE
```