Why wouldn't someone accept Gentzen's consistency proof?

Solution 1:

Essentially due to Gödel's Incompleteness Theorems, proofs of the consistency of $\mathsf{PA}$ must involve methods that transcend $\mathsf{PA}$ itself. If one has any doubts about the consistency of $\mathsf{PA}$, those doubts are likely only to be amplified concerning the methods used to prove the consistency of $\mathsf{PA}$. (For example, from $\mathsf{ZF}$, then you can easily construct a model of $\mathsf{PA}$, but the consistency of $\mathsf{ZF}$ is "more debatable" than that of $\mathsf{PA}$, so you haven't really gained anything.)

Gentzen's proof relies on infinitary processes (in particular, induction up to $\varepsilon_0$; see Wikipedia), and may not have been accepted by the Hilbert school (who sought purely finitary proofs of consistency). The ordinal $\varepsilon_0$ is important here because (assuming its consistency) $\mathsf{PA}$ cannot prove that it is well-founded, and it is basically this move that transcends $\mathsf{PA}$.