Lose the sprint, win the marathon

Clearly, if Chris lost to any other player in all $21$ races, he could not have taken $1$ st place overall, because the other player would have been faster than he was. We claim that his worst possible position is $188$ th.

If he ranked $188$ th in each race, it is possible for him to have lost by only $1$ second to the $187$ faster competitors in each race, but be faster than all of the other $10$ by hours. Since he was faster than $10$ competitors in each race, and there are $21$ races, it is possible for him to have been faster than all of the other $197$ competitors in at least one race. Note that, since he only lost by a second when he lost and won by hours when he won, his overall time would still be faster than the other $197$ competitors, because his margin of winning is more than his margins of losing combined.

If he ranked $189$ th in each race, he would've been faster than $9$ competitors in each race. Since there are only $21$ races, and $9\times21=189<197$ , by Pigeonhole, there must exist a player who was faster than Chris in all races. Thus, it would've been impossible for him to have the fastest overall time.

The final answer is thus $\boxed{188}$ .