MiniDSP SHD. I fully believe that some of the really, really, really good gear could do a lot better on it, but I am talking Theoretica and such.
As for mixing and such, there are a lot of issues with that as an argument.
1. Some of the really best recordings don't have it, but that is a very limited volume. Ex Octave's stuff and DSD from old tapes
2. Mixing and DSP are very different. Level mixing is simple. Parametric equalization is sometimes used in mixing but means the recordings had issues. Parametric timing changes are not done much at all in mixing. The latter two are in DSP, and are very complex and easy to screw up.
3. All the digital manipulation everywhere in the chain tends to introduce errors, and it adds up. I think a good analogy is freezing stuff. Sure - the fish you buy at the store has often been frozen before. But that does not mean that freezing it again at home and thawing it again is not going to make it worse.
You mentioned things can add up. I agree and also stipulate they add up quicker when done in a haphazard way.
Math and processing has come a long way. Math does not suffer from parts tolerance variances and age degradation.
Math and processing potential improves with age as history has proven.
I experimented with math and have a few examples below.
The best measuring DAC today is @ 130dB so to drive that DAC to its potential you need something cleaner than its maximum potential.
DSP can be done today in 16, 32, 64 and 128-bit, yet many tools chose lower precision math.
Here are some examples that I have tested:
Here are 4 FIR XO's processed in 32-bit ( 16-bit has a higher noise floor than -130dB so is already under-performing the DAC )

Note the noise floor of -140dB. Just marginally better than the best contemporary DAC.
Here are the same 4 FIR XO's processed in 64-bit.

Note we are now down to @ -240dB, surpassing the best contemporary DAC's capability.
Now look at the same 4 FIR XO's processed in 128-bit.

Note we have now exceeded the precision of the display tool.
It is possible with today's tools to do all the math and convolution in 128 and 64-bit until the DAC's doorstep.
I would suggest the difference between good and poor DSP performance today is the choice and consistency of the filter and processing chain (using your favorite DAC du jour).
I have found some software tools to cut corners, thus avoid them accordingly in favor of ones that do not.