Having a static object that is used as a lock typically is not desirable because only one thread at a time in the whole application can make progress. When you have multiple classes all sharing the same lock that's even worse, you can end up with a program that has little to no actual concurrency.
The reason Java has intrinsic locks on every object is so that objects can use synchronization to protect their own data. Threads call methods on the object, if the object needs to be protected from concurrent changes then you can add the synchronized keyword to the object's methods so that each calling thread must acquire the lock on that object before it can execute a method on it. That way calls to unrelated objects don't require the same lock and you have a better chance of having code actually run concurrently.
Locking shouldn't necessarily be your first go-to technique for concurrency. Actually there are a number of techniques you can use. In order of descending preference:
1) eliminate mutable state wherever possible; immutable objects and stateless functions are ideal because there's no state to protect and no locking required.
2) use thread-confinement where you can; if you can limit state to a single thread then you can avoid data races and memory visibility issues, and minimize the amount of locking.
3) use concurrency libraries and frameworks in preference to rolling your own objects with locking. Get acquainted with the classes in java.util.concurrent. These are a lot better written than anything an application developer can manage to throw together.
Once you've done as much as you can with 1, 2, and 3 above, then you can think about using locking (where locking includes options like ReentrantLock as well as intrinsic locking). Associating the lock with the object being protected minimizes the scope of the lock so that a thread doesn't hold the lock longer than it needs to.
Also if the locks aren't on the data being locked then if at some point you decide to use different locks rather than having everything lock on the same thing, then avoiding deadlocks may be challenging. Locking on the data structures that need protecting makes the locking behavior easier to reason about.
Advice to avoid intrinsic locks altogether may be premature optimization. First make sure you're locking on the right things no more than necessary.