It seems to me that Gerhard Gentzen's sequent calculus could just omit negation and falsehood, and still prove any classical tautology in a suitable form. (For a specific formula, falsehood gets replaced by the conjunction of all relevant propositional variables. For predicate logic, falsehood gets replaced by the conjunction of the universally quantified predicate symbols, including for example $\forall x\forall y\ x=y$. One could also introduce a constant $F$ and axioms like $F\to\forall x\forall y\ x=y$. More details about the consequences of removing falsehood can be found here.) This doesn't seem possible for his natural deduction calculus, where the law of excluded middle is used to arrive at classical logic.
Why hasn't he adapted the relevant rule from his sequent calculus to his natural deduction calculus: $\begin{array}{l} A\to(B\lor C) \\ \hline (A\to B)\lor C\end{array}$
At first sight, this rule doesn't look worse than $\begin{array}{l} \\ \hline A \lor \lnot A\end{array}$ or $\begin{array}{l} \lnot \lnot A \\ \hline A \end{array}$, and it would have mirrored his sequent calculus more closely. Of course, in the sequent calculus he didn't need to write $\lor$, and this made this deduction rule look even more attractive. But being able to omit both negation and falsehood seems attractive to me, independent of how attractive the rules themselves appear.
The only reason I could come up with is that he developed his natural deduction calculus first, became dissatisfied with it, then developed his sequent calculus, and didn't find it important to further improve his natural deduction calculus, because it had other irreparable flaws anyway.