Always use new token issuing flow

What does this MR do and why?

Always use new token issuing flow.

This is a refactor. We migrated a set of unit primitives to the new Token Issuing interface for Cloud Connector.
Now we want to use this flow by default.

This MR also cleans up corresponding feature flags as they were rolled out:

• code_suggestions_new_tokens_path: #560317 (closed)
• anthropic_proxy_new_tokens_path: #561289 (closed)

How to set up and validate locally

This is no-op change.
But for safety, we can ensure Code Completion works as expected (we need to run as SaaS).

MR acceptance checklist

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Related to #561289 (closed)