Why shouldn't Java enum literals be able to have generic type parameters?

This is now being discussed as of JEP-301 Enhanced Enums. The example given in the JEP is, which is precisely what I was looking for:

enum Argument<X> { // declares generic enum
   STRING<String>(String.class), 
   INTEGER<Integer>(Integer.class), ... ;

   Class<X> clazz;

   Argument(Class<X> clazz) { this.clazz = clazz; }

   Class<X> getClazz() { return clazz; }
}

Class<String> cs = Argument.STRING.getClazz(); //uses sharper typing of enum constant

Unfortunately, the JEP is still struggling with significant issues: http://mail.openjdk.java.net/pipermail/amber-spec-experts/2017-May/000041.html

The answer is in the question:

because of type erasure

None of these two methods are possible, since the argument type is erased.

public <T> T getValue(MyEnum<T> param);
public T convert(Object);

To realise those methods you could however construct your enum as:

public enum MyEnum {
    LITERAL1(String.class),
    LITERAL2(Integer.class),
    LITERAL3(Object.class);

    private Class<?> clazz;

    private MyEnum(Class<?> clazz) {
      this.clazz = clazz;
    }

    ...

}

Because you can't. Seriously. That could be added to the language spec. It hasn't been. It would add some complexity. That benefit to cost means it isn't a high priority.

Update: Currently being added to the language under JEP 301: Enhanced Enums.