Honestly, I found the algorithm description cryptic. It just may be me not "in the know" when it comes to details, with your "bunch of knowledge [that] exists in industry only" being the details the authors didn't care to elaborate on in the algorithm description. Maybe a part of the reason for the failure to replicate is that other people found it cryptic as well and didn't understand crucial details.

BTW, in your opinion, is there any readable text on the LK algorithm? I'm afraid that all I've read so far seems to be suffering from this problem. I have a very specific optimization need that doesn't seem be covered even by LKH-3 (it seems to be that my problem could be treated either as asymmetric TSP with time windows on a small number nodes and a non-metric distance matrix, or alternatively from the other side as VRP with asymmetric distances, unknown number of vehicles, and bounded trip duration for every vehicle, but either of the two needs an additional constraint that all vehicle trips should get close to the number of working hours within a day with the exception of one trip which can be shorter - basically I need an "ATSP with sleeping breaks") so I may need to roll out my own implementation of something and for an outsider, there doesn't seem to be a good text on implementing these things realistically without having some prior knowledge already.