Are esprit de corps runs still relevant? If yes, at what echelon?

MAJ (Join to see) - As for the Esprit de Corps Runs? I would ask whose? This doesn't answer the key questions of : Will your soldiers follow you?
Unfortunately, like most intangibles, a unit's esprit de corps not be readily apparent, BUT its absence is obvious.
The lack presents in new soldiers not yet integrated into the unit and units that have not been bloodied (together) when facing new circumstances.
Even veteran elements that are 'tired' can have a lapse upon facing a sudden crisis. History has shown that veterans that face a slow, gradual increase in 'pressure' often hold out well beyond command expectations. Bastogne, Pusan, Heartbreak, 73 Easting and others are the best example of conventional war actions. There are thousand of example where small element have fought where situations (sometimes won, sometimes lost) have been key for a larger unit's success.
Planning physically rigorous team activities and mentally stressful chain of of command (including NCOs) exercises are a major way to train and build Esprit de Corps. You have an end game, accept failure, praise the best efforts. promote self-honesty and put forth realistic goals at each level of development.
I hope this my comments helped explain my observations.

My final is to say that these types of runs are another, inexpensive and easily applied 'Gage' for the commander.

I liked them at BN level. I was the company guidon bearer for a good part of my time during active duty. Because we were the HSC, we ran in front, right behind the BN command staff. That meant I was usually the one to initiate the run-around-the-formation challenges (or often my CO turning his head back at me saying "go!"). I was a much better runner then, and I loved the looks of hatred from the other companies' bearers when I ran past and they knew they had to meet the challenge.

The other bonus, in my opinion, was it gave me a chance to stretch out a bit from the horrid pace...even as the lead element.

I believe they are. When I was at my last base, we had Group PT every Wednesday. Sometimes it included runs and sometimes it was brutal cross fit type training. But everyone did it. 6 Squadrons formed up and every week it was led by one of the squadrons, they rotated. You fell in formation wherever you were, not by squadron. It showed the importance of being a unit from top down. It put people together that never saw each other because they have different AFSC's and worked in different facilities. They got to know each other and what each other brought the fight. It built networking, esprit de corps, morale etc. People actually looked forward to it from the senior leaders down.

Back when I was stationed at Ft Carson,once a year we did Division runs,the whole Post,I liked them but an all day affair,I did not care for BDE runs .We were singing cadence about the Col,he diverted from pavement to freshly snowed on tank trail,while in boots,nothing like few extra pounds of mud.

They give me a chance to chat with the BSLs from other units. We tend to stroll more as well, in order to avoid passing the formation...