How Come the Low Pass Filter in Sobel Operator Isn't Normalized?
The answer is simple, the Sobel Filter is a composition of Lows Pass Filter (LPF) and High Pass Filter (HPF). The composition is done by convolution.
Now, indeed the LPF presented above $ {\left[ 1, 2, 1 \right]}^{T} $ has amplification in the DC value (Its sum is 4 so the amplification is 4). Yet it is convolved with an HPF filter which rejects the DC component.
Convolution is multiplication in the Frequency Domain, since we multiply 4 by zero we essentially get zero.
Actually multiplication of LPF and HPF gives a Band Pass Filter (BPF) (In case they have some overlap in Frequency Domain). Hence in the case above, the Sobel Filter is actually a BPF.