I am studying Rational Points on Elliptic Curves by Silverman and Tate. They only prove the weak Mordell-Weil Theorem in the case when the curve $y^2=f(x)$ has a point of order $2$, i.e., $f(x)$ has a rational root. This means $f(x)$ is reducible, so I suppose it's a quite special case. The authors say the general case needs some knowledge about ideal class group and units. However all the proofs I can find (for example the proof in the other book by Silverman) seem to also use group cohomology or something similar.
Is there a proof that resembles the proof in Silverman and Tate? Namely, if $\alpha$ is a root of $f(x)$ and we view $E:y^2=f(x)$ as a curve over $K=\mathbb{Q}(\alpha)$ (so that we have the $2$-isogeny), can we define a suitable map from $E(K)$ to $K^*/(K^*)^2$ and show that $E(K)/E(K)^2$ is finite? Maybe instead of $K^*$ we should use the group of fractional ideals? It seems like the same proof goes through in case $K$ has class number $1$, although that may be rare.
The authors motivate the $2$-isogeny from $E:y^2=x^3+ax^2+bx$ to $\overline{E}:y^2=x^3-2ax^2+(a^2-4b)x$ using uniformization by Weierstrass $\wp$ function; it ultimately comes from cutting the fundamental region of the lattice in half. But since lattices correspond bijectively with elliptic curves in Weierstrass normal form, shouldn't this work for all elliptic curves, not just those with rational roots?