This explanation neglects one aspect that's key to the design. The diff amp is not just subtracting the low-pass output from the input. The arithmetic performed by the diff amp involves all four op-amp integrator outputs. This completely changes things. One reasonable way to analyze it is to start from the low-pass output and work backwards through the four integrators. For each integrator, going from its output backwards to its input adds a 6dB/oct positive slope to the frequency response that's ideally independent of frequency. Working backwards in this way gives the following results:
IC2a output: Low-pass with 24dB/oct high frequency rolloff
IC3b output: Bandpass-like circuit with 6dB/oct low frequency rolloff and 18dB/oct high frequency rolloff
IC2b output: Bandpass filter with 12dB/oct low frequency rolloff and 12dB/oct high frequency rolloff
IC3a output: Bandpass-like circuit with 18dB/oct low frequency rolloff and 6dB/oct high frequency rolloff
IC1b output: High-pass with 24dB/oct low frequency rolloff
In terms of the electrical performance, the state variable filter works a lot better than one would expect from looking at how many op-amps are needed. That's because all the feedback loops are negative feedback, reducing the distortion to lower levels than one would expect. Care has to be taken with the transfer function from the input to each of the op-amp outputs. These should be scaled so that no op-amp clips before the output does. It turns out these can be scaled in such a way as to not affect the transfer function from input to each desired output.
Also, the Sallen-Key filter electrical performance is much worse than one would expect. This is because the feedback path that makes the filter work is pure positive feedback. This was studied by a guy named Billam in his 1978 AES article "Harmonic Distortion in a Class of Linear Active Filter Networks" (vol. 26, #6). How much worse is the distortion of the overall filter than that of the buffer itself? He did an example of a filter with cutoff frequency 1940Hz and measured what he called the "distortion aggravation factor" (ratio of filter distortion to that of the buffer alone). Here's his table.
You can see that the distortion of the filter near the cutoff frequency is more than 80x that of the buffer itself. Not good. It's one case for which design minimalism yields poor results.
