The trick is that, even though the augmented state is uncontrollable, you still use it in the predictions, so the MPC algorithm can still compensate for it. Take a look at the before-last graph in the paper, see how that technique improves predictions after learning the real-time disturbances.
The code is C++ code, It could be ported to an MCU if there is a fairly recent C* compiler (C11+). But the algorithms are quite expensive, there are multiple matrix inversions, the size of the matrices been as large as the data you put in into the tool. typically between 500 and 10000 data points.
You are right, but the problem of a possible analogous "Machine Learning Theory" is that machine learning problems are highly nonlinear, and control theory has only been "solved" for linear problems. It we could solve nonlinear control that could immediately be applied to machine learning.
“Linear systems are important because we can solve them.” — Richard Feynman
The problem with many adaptive control techniques is that is very difficult to prove the closed loop stability and performance margins, because such systems are highly nonlinear and we do not have yet the mathematics to provide simple guarantees (for example, I would not get into an aircraft where the control systems have not been rigorously, mathematically, designed)
The integral problem of LQR and MPC is indeed not the first topic discussed on textbooks, but there has been a solution for that for year, called "disturbance model". You you can have an "adaptive MPC" following some design procedures. Check out this reference: