Regular language concatenation with superset
Simply put, this statement of yours:


  On the other hand $\tilde{L}$ seems to contain strings E does not, i.e. those
  that are formed by concatenating non-empty string from both languages.


is wrong.

As you say, $E$ contains every string containing only symbols from $A$. Since $L$ also only contains strings with symbols from $A$, $\tilde{L}$ must as well. Concatenating strings that have symbols only from $A$ will just get you another string whose symbols only come from $A$.

Your statement is flipped; it is $E$ that contains strings that $\tilde{L}$ may not.