Is there a reasonably simple proof that the normal distribution is the unique sum-stable distribution with finite variance? That is, for the claim
Claim: If $X_i$ are i.i.d. with mean zero and finite variance and $X_1 + X_2 \overset{\text{law}}{=} \sqrt{2}X_3$ then $X_i \sim \mathcal{N}(0,\sigma)$.
A natural place to look for it would be in a book on sum-stable distributions, for instance "Chance and Stability" by Zolotarev et al. But it just mentions that this follows from the central limit theorem.
One can prove this after proving the central limit theorem, but I'd rather motivate the central limit theorem using this simpler fact.
My secret motivation is that I find proofs of the central limit theorem unsatisfying. (Even after scouring Stack Exchange and seeing proofs in terms of eigenfunctions of operators, strictly increasing entropy, dynamical systems, etc.) I just think there should be a more intuitive way.