Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it implies that only finitely many torsion subgroups are possible, which seems like a much weaker result.

My question: Is there any way to see that the weak version of bounded torsion (only finitely many groups occur, but nevermind what they are) is true, without recourse to the full proof of Mazur’s theorem? I saw some references to a paper of Demjanenko from 1971 (EDIT: originally said 1975) which claims to prove this, but it’s only available in Russian and other sources don’t seem to think its argument is correct.