The issue with
List<String> list = new LinkedList();
is that on the left hand side, you are using the generic type List<String> where on the right side you are using the raw type LinkedList. Raw types in Java effectively only exist for compatibility with pre-generics code and should never be used in new code unless
you absolutely have to.
Now, if Java had generics from the beginning and didn't have types, such as LinkedList, that were originally created before it had generics, it probably could have made it so that the constructor for a generic type automatically infers its type parameters from the left-hand side of the assignment if possible. But it didn't, and it must treat raw types and generic types differently for backwards compatibility. That leaves them needing to make a slightly different, but equally convenient, way of declaring a new instance of a generic object without having to repeat its type parameters... the diamond operator.
As far as your original example of List<String> list = new LinkedList(), the compiler generates a warning for that assignment because it must. Consider this:
List<String> strings = ... // some list that contains some strings
// Totally legal since you used the raw type and lost all type checking!
List<Integer> integers = new LinkedList(strings);
Generics exist to provide compile-time protection against doing the wrong thing. In the above example, using the raw type means you don't get this protection and will get an error at runtime. This is why you should not use raw types.
// Not legal since the right side is actually generic!
List<Integer> integers = new LinkedList<>(strings);
The diamond operator, however, allows the right hand side of the assignment to be defined as a true generic instance with the same type parameters as the left side... without having to type those parameters again. It allows you to keep the safety of generics with almost the same effort as using the raw type.
I think the key thing to understand is that raw types (with no <>) cannot be treated the same as generic types. When you declare a raw type, you get none of the benefits and type checking of generics. You also have to keep in mind that generics are a general purpose part of the Java language... they don't just apply to the no-arg constructors of Collections!