To this day the question I have about the Quadratic Seive is something like "what is being reduced in degrees of freedom by each relation added to the matrix?"

It seems like the sequence of primes in the factor base (which are quadratic residues) is very much related to the residues of the factors (they are) and somehow this it's doing in the abstract something like CRT to construct the factors. But I have never read a deeper explanation beyond the mechanics of implementation.