LKH does support a lot of things mentioned, but for practical usages it would not work. It's nice to leave it running and see what can be accomplished but asking it to give you back something in 1 second, with a lot of constraints, gives back solutions that are not feasible.

In the basic TSP there is a lot of data.

For example, the reason why minimum spanning tree works is because the algorithm makes use of the relationship between vertices. Similar techniques use alpha-nearness, Steiner trees and direct modifications of distance matrix to create relaxations of the TSP and improve the performance of local search (I believe most are implemented in LKH).

I am obviously not expecting NNs to be capable of doing something like that currently but I'm hoping they might be able to discover interesting instance patterns for something more constrained.