Rollout new Engine API client on Teku

[lucassaldanha]

Feature implementation: #10875

This issue has the follow-up tasks required to roll out the new engine API as the new default.

Here is the plan:

1. We'll update the flag --Xnew-engine-api-client-enabled to default to true in our next release (26.7.0)
2. After the change is merged, we can deploy develop to all nodes and monitor for any abnormal behaviour
3. If we are happy with the results, we can proceed to implement Remove --Xnew-engine-api-client-enabled flag #10878, removing the flag and the old code