I will try to answer on your question with this: Least Square equalizer is minimizing the intersymbol interference of your signal (ISI), between the symbols, which does not translate directly to minimization of the bit error rate. When I am saying 'directly' that means that there is no exact linear dependency between the error reduction in the equalizer, and the bit error rate. However, they generally tend to go into the same direction.
The reason for that is the nature of equalizer, as the nonlinear estimator.
The chain of processing that you have shown consist of processing modules, and some of them are nonlinear, which interferes with the bit error rate minimization. In other words, bit error rate minimization is not a criterion that is used by the equalizer.
Below is the mass of papers that are talking about it, and it is not a simple matter.
Please note the first one: "Approximate Minimum Bit-Error Rate Equalization for Binary Signaling" which actually argues exactly the same line of inquiry that you are advocating: "Although most linear and decision-feedback equalizers are designed to minimize a mean-squared error (MSE) performance metric, the equalizer that directly minimizes biterror rate (BER) may signiﬁcantly outperform the minimum MSE equalizer".
This means that the digital bit error based criterion would be a better choice for reducing the bit error rate than the one that is typically used.