I admittedly don't know anything about Halo but what are the map size like in the game because I don't necessarily see Overwatch as a good comparison. There is no "sprint button" for people other than Soldier 76 but a large number of characters have mobility options made available to them.
Genji, Tracer, Junkrat, Widow, D. Va, Rein, Winston, Lucio, and Mercy have the ability to cover ground quickly though Mercy has the quirk of needing someone target. Pharah can fly, Sombra can teleport huge distances, Reaper has his wraith form, McCree can roll and Zenyatta gets a speed buff when he uses his ult. Many of these abilities can be used to get back into the fray quicker (some clearly better than others) or be used in combat situations itself.
Going back to the fact that I know nothing about Halo, maybe whether or not Overwatch is a good example doesn't matter. If the structure of Halo is smaller maps with more action or the game sufficiently provides you with mobility options either through vehicles, teleporters and what have you, then a sprint ability might not be necessary.