 Using the angular uniform in the Box-Muller radius breaks normality

If $V$ is uniform on $(0,1)$, the displayed random variable is not standard normal. For $V\le 1/2$ it is nonnegative; for $V>1/2$, its radius is at most $\sqrt{2\log 2}$, so it is always bounded below by $-\sqrt{2\log 2}$. A standard normal distribution has an unbounded lower tail.

The Box-Muller transform instead uses a common radius $\sqrt{-2\log U}$ independent of the angle $2\pi V$ in both Cartesian coordinates. Their polar density is $r{e}^{-r^2/2}/(2\pi )$, and dividing by the coordinate Jacobian $r$ gives the product of two standard normal densities. Changing a logarithm's uniform input changes the distribution, even if the resulting expression superficially resembles that formula.