Surely the point of the exercise is to find a horse, and the biggest difficulty is what you do with it when you actually catch up.

And now consider this from the perspective of a wild horse: a strange being approaches, your instinctual neophobia tells you to flee. At a safe distance, you stop, only for the being to arrive again and the cycle to continue. You sleep, only to be awoken by its approach. Finally, too exhausted to run any further, you wait for the inevitable… only for the being to feed you a carrot, brush you mane, and wander off.

I do not think horse would just "wander off" after that. It would need veterinary treatment and a long time for recovery.

Look up details for horse marathon. Horses need several breaks and veterinary supervision. And that is just a few miles!

That was the human wandering off, not the horse.