Simplify and Fix Synchronization in InternalTestCluster (#39168) (#39241)

* Simplify and Fix Synchronization in InternalTestCluster (#39168)

* Remove unnecessary `synchronized` statements
* Make `Predicate`s constants where possible
* Cleanup some stream usage
* Make unsafe public methods `synchronized`
* Closes #37965
* Closes #37275
* Closes #37345

Author: original-brownbear