What is the importance of disjoint edges in k-opt algorithms?
I have a partial answer to my own question. I believe there is a benefit to considering adjacent edges. The full power of the heuristic is not achieved when only disjoint edges are considered.

To arrive at this conclusion, I implemented two versions of the 4-opt algorithm in C#. One version considers only disjoint edges and the other considers both disjoint and adjacent edges. Then, I used both algorithms to solve the Travelling Salesman problem for 1000 scenarios of 25 nodes each.

The two algorithms found the same solution 52.7% of the time.
The disjoint+adjacent algorithm found a better solution than the disjoint-only algorithm 45.7% of the time.
The disjoint-only algorithm found a better solution than the disjoint+adjacent algorithm 1.6% of the time.

The disjoint+adjacent algorithm found tours that were, on average, 99.1% of the length of the tours found by the disjoint-only algorithm.

Based on these results, I'm fairly confident that there is a benefit to exploring adjacent edges in the k-opt algorithm.

I call this a partial answer because I'm not yet clear on whether it is efficient to explore adjacent edges. Yes, exploring adjacent edges in my experiment improved the tour length by an average of 0.9%, but perhaps I could have improved the tour length more than that by using the same computing effort exploring some 5-opt steps instead.

My experimentation suggests that it is optimal to first explore 2-opt steps, then to explore 3-opt steps that include adjacent edges, then to explore 3-opt steps that include no adjacent edges, then to explore 4-opt steps that include adjacent edges, then to explore 4-opt steps that include no adjacent edges, etc. However, that is preliminary speculation. If anyone knows of k-opt documentation that covers this level of detail, please let me know about it.