I assume that one can also derive some atomic statements from
<E-STMT>, otherwise the language of both grammars would be empty and all would be quite trivial. :-) But that aside:
The new grammar should do two things: it should be unambiguous and it should generate the same language.
<E-IF-THEN-ELSE> are introduced to remember part of the context in which the variable occurs. What you want to accomplish, is that the variables
<IF-THEN-ELSE> cannot occur in a position directly preceeding an
else symbol, and also that the variables
<E-IF-THEN-ELSE> must occur in a position directly preceeding an
else symbol. So, if we see a
<E-IF-THEN-ELSE> somewhere, we know that the next symbol must be
else, and if we see an other variable we know that the next symbol cannot be an
else. This way we have some week form of "context-awareness".
If we remove the rule for
<E-IF-THEN-ELSE> and replace the rule for
<E-STMT> -> <IF-THEN-ELSE>
then we can derive the following:
if condition then <E-STMT> else <STMT> $\to$
if condition then <E-IF-THEN-ELSE> else <STMT> $\to$
if condition then if condition then <E-STMT> else <STMT> else <STMT>
and here a
<STMT> variable which directly preceeds an
else symbol. This will lead to ambiguities.
If we try to fix this and additionally change the if-then-else rule to
<IF-THEN-ELSE> --> if condition then <E-STMT> else <E-STMT>
then we can perform the derivation
if condition then <E-STMT> else <E-STMT>
and here we have an
<E-STMT> variable without a following
else symbol. Since the
<E-STMT> generates no if statements without else clause, the language of the new grammar is not the same.
The grammar which was proposed in the book solves both these problems.