Yes, this is indeed a missing piece of the argument.
The 'real' argument is to just derive the Lorentz transformations from first principles, then note that $y' = y$ for a boost along the $x$ direction. But this isn't pedagogically useful, because usually people want to see proofs of time dilation and length contraction individually first, so they can accept the Lorentz transformations in the first place. So we shouldn't use this powerful tool.
Instead, here's a simple 'physical' argument. Suppose I draw a horizontal red line along a wall, one meter above the ground. Then I take a meterstick, put one end on the ground, put a blue paintbrush on top of it, and move it extremely fast horizontally.
If transverse length contraction exists, then the blue line will end up below the red line. But we could also have done this experiment in the frame of the meterstick, which sees the wall moving, in which case the red line will be below the blue line. These can't both be true, so the only possibility is that transverse length contraction doesn't occur and the lines are on top of each other.
I quietly slipped in an extra assumption above: that by symmetry, the transverse length contraction due to velocity $v$ should be the same as that due to velocity $-v$. Relaxing this gives generalizations of the Lorentz transformations described here where transverse length contraction does exist.
In practice, we justify this assumption using the observed symmetry of the universe; it is a hidden but necessary extra axiom of special relativity.
