Never heard of “At Ease” march, but I have given the command “Route Step, March!” It is as you said, to break the rhythm.
How you can march a formation at ease leaves me at a loss.
Apparently, at least since 2006, you can. From the web:
The command Route Step, March permits talking while marching. Talking and movement are permitted as long as dress, cover, interval, and distance are maintained. The command At Ease, March is similar except silence is maintained.Aug 21, 2006
Per FM 3-21.5: “upon the command, the soldier is no longer required to retain cadence; however, silence and approximate interval and distance are maintained.”
I don’t remember using it very often...and never in the ‘real army’ after West Point...but I could probably count on one hand the number of times we ‘marched’ in the regular army, outside of PT running formations. And I don’t really know why it was used, other than laziness. In this picture, I know they are walking up a route that has long steep grades...some of the women are in heels...and for some reason they are only using half the road and four across (possibly elongating the unit and making it hard to hear somebody calling cadence). So, the short timer cadet calling cadence probably just decided to say ‘screw it’, and called at ease march.