How can any spectral analysis show that Sigma-Delta ADC reduces quantization noise?
Any quantized signal has a certain minimum quantization noise level, which is related to the number of bits per sample as you have noted. Even a delta-sigma converter cannot reduce the noise below this theoretical floor.

The noise reduction (noise shaping) that is talked about in these converters relates to the fact that they use a very coarse quantizer (very often a single comparator yielding just one bit a time), which has a very high quantization noise level at its output. This noise is uniformly distributed throughout the spectrum defined by its sample rate (which is very high).

It is the function of the noise-shaping circuitry to shift as much of this noise energy as possible to a  frequency band outside the band occupied by the signal of interest, in order to get as close to the minimum noise level in the passband as possible. The noise is then removed by digital filtering. At the same time, the sample rate is reduced to the desired output sample rate (decimation).

Ultimately, a very noisy string of high-speed 1-bit samples is converted to a nearly-optimal string of N-bit samples at a lower speed. The key advantages of this approach are that it's possible to get both high resolution and high linearity with relatively low cost &mdash; properties that are particularly valuable in applications such as digital audio.