I feel like sprinting in games is more psychological rather than utility. In games where there is no stamina bar the developers could just make that your normal moving speed but then players will want to move faster still. So you put in a sprint button and instantly what feels like a long walk from one point to the next is a brisk jog.

Its the same psychology behind waiting. Let us say you're going to the store and there is a long wait. As long as you're in a line and its moving every so often the wait doesn't feel bad to you. But the moment you remove the line and replace it with an equal amount of time to wait with no indication how long it will take people to become impatient and irritable. They're waiting the same amount of time, but in the first example they 'feel' like they're waiting less time because they can see the queue moving along.

As for games with stamina bars, unless running gives you a big advantage in combat I find it pointless to tie running down to a stamina bar, especially when I'm out of combat. Give me infinite stamina when I'm out of combat so I can just move around faster.