Why does Feynman say “That is all there is to the theory of relativity – it just changes Newton’s law by introducing a correction factor to the mass”

The claim is wrong because Newton's second law does not become $F=m \gamma a$.

Relativistic kinetic energy $$K_{rel}=mc^2\left(\frac{1}{\sqrt{1-(v/c)^2}}-1 \right)$$ is not obtained from [Newtonian] kinetic energy $$K_{nr}=\frac{1}{2}mv^2$$ replacing $m$ by $m\displaystyle\frac{1}{\sqrt{1-(v/c)^2}}$.