Imagine that $k$ runners on a $1$ mile long circular running track start
running with different constant speeds. Pick one of the runners. Will
there be a time when this runner is $1/k$ miles away from all other

The runners might need to run several circles before this happens.

Imagine that you participate in a three person race, and you are running at $7$ mph while the others are running at $6$ mph and $8$ mph. Then after half an hour you have run three-and-a-half laps, so you are in the middle of the track, while the others have run three and four laps respectively. So both of them are at the starting line. Your distance from them is half a lap, which is bigger than $1/3$ lap.

If there are $k$ runners, will you eventually be $1/k$ miles away from them? The only thing you know about speeds is that all the runners' speeds are different, and each runner maintains the same constant speed for ever.