I would not say that no one should ever do linear combination fitting for EXAFS.  For sure, linear analysis of XANES is quite robust and verified many times to give good results, at least at level of a few percent.  Linear analysis of EXAFS suffers more data processing challenges and conceptual problems that limit its robustness.  For sure, there are cases for which it can work well. 

Longer answer: 

Any linear analysis (LCF, PCA, MCR-ALS, etc) of XANES works reasonably well (typically to a few percent) because:

   a) the processing needed is minimal.  Data need to have a common energy calibration better than the intrinsic energy resolution -- typically energy calibration of 0.25 eV or better will be OK.  Data need to have a consistent normalization of mu(E), typically to a few percent.   Variations in these processing steps will have a direct and negative effect on the results.

   b) conceptually, the assumption is that there exists a nearly 1 to 1 correspondence between "local chemical configuration" and "measured XANES", and that the "local chemical configurations" that are being investigated are discrete and well-defined (ie "iron carbonate") and not continuous.   That is, if you determine that your Fe XANES spectra is "50% iron carbonate and 50% iron sulfate" then implicit conclusion is that 50% of the iron atoms are iron carbonate and 50 percent are iron sulfate, not that all irons are 50% carbonate and 50% sulfate. 

To be clear, linear analysis of XANES does not work well to ppm levels, partly due to the poor experimental contrast (that is, mu(E) tend to all look alike and features are intrinsically broadened to the ~eV level), but also conceptually, because at the ppm level, local chemical configurations are not always limited to 3 to 10 discrete states.

Linear Combination EXAFS is more challenging from both the processing and conceptual point of view.  

For Processing, EXAFS requires more data processing than XANES.  The selection of E0 and the background mu0(E) will have an effect on linear analysis of EXAFS if not done consistently.  It is not really obvious how E0 or mu0(E) can be selected consistently for very different spectra.

Conceptually, EXAFS is much more sensitive to disorder and subtle variations in the bond lengths (thermal or static disorder) and can have significant variation in its sensitivity to second and further neighbors.   In that sense, EXAFS is much less discrete and much more continuous in its variability across different kinds of local structures.   

Again, this is not to say that linear analysis of EXAFS cannot ever work, just that is probably more limited in applicability and absolute accuracy than linear analysis of XANES.  Of course, for EXAFS you can also do an actual fit of structural parameters.  The information content is somewhat limited so that refining multiple overlapping components may not always be possible, and linear combinations of end-member spectra may look attractive....

Hopefully, anyone who has other insights or experiences will be able to correct any of my misunderstandings.