Without solving it, is there an elementary way to show that $X^3+Y^3=Z^3$ has a finite number of primitive [and non-trivial] integer solutions?

Question

Considering the cubic case of Fermat’s Last Theorem, I make the following claim:

Proposition: The Diophantine equation $$ X^3 + Y^3 = Z^3 \tag{$\star$} $$ has a finite number of primitive [and non-trivial] integer solutions $(x,y,z)$.

Is there a simple and elementary way to prove this statement?

Note: I am aware of Wiles’s proof of FLT in the general case, and the infinite descent proof of the cubic case by Euler et al., and the Mordell–Weil theorem, etc. I’m just curious, independent of those things, whether this weaker proposition has a simple and elementary proof.