This question is motivated by my impression, that finding better heuristics for the TSP problem (or any other $NP$-complete problem) is "only" of practical interest, but doesn't provide any progress towards a decision of the $P=NP$ question.
The reason for my impression is that all heuristics I'm aware of, do not have the same "invariants" that optimal tours have, most prominently the invariance of the solution set of edges if the problem instance is subjected to the addition of vertex weights.
The Christofides heuristic for example had the best theoretical upper length-bounds for the tours it generates for at least two decades and still is better than the best tour expansion heuristics, IMHO there is nothing that can be concluded from that difference in performance, that would help to understand, why $NP$ problems are so hard.
is my impression right or are there examples of heuristics that are considered to have contributed to progress in answering the $NP=P$ question, because they are based on newly discovered provable properties of optimal tours?
I am looking for theorems about the $NP=P$ problem that made, resp. make, use of (then) novel ideas of TSP heristics or, were motivated by such ideas. I kindly would ask to abstain from expressing opinions about subject; the importance of research on better heuristics is unquestionable to me!