Except if you were an exceptional runner, you could choose to not make move 1 until you have 5 minutes left. Because you have 60 minutes that would give you 11 rounds of the 800m race. So you can see how a good runner could simply chose not to play a move and win anyway. Of course it wouldn't have to be this obvious. Also 1 more before mate you could chose to use up your remaining races by waiting 5 minutes etc.
This is one reason why winning the race by a minute shouldn't be enough to declare an outright winner -- I think possibly a time penalty on the chess clock would be best.
Also running doesn't fatigue your chess playing ability -- at least not like getting knocked in the head would. Physical fatigue would diminish the chess ability very little IMO.
Also boxing may be violent, but it's controlled and consensual. I don't agree with the violence being "definitely not good" as you describe it.
Seeing that Chessboxing article gave me an idea. I like the concept of chess boxing, combining physical fitness with mental intelligence. But it encourages violence, which is definitely not good. I propose a different alternative "Chessrunning" which combines physical fitness with mental intelligence, much like Chessboxing, but without the violence. Here are the rules that I suggest:
First there would be a 5 minute round of chess, with time controls of 60 0. Then there would be an 800m running race. If anyone wins by a margin greater than 1 minute then that person would be claimed winner. Then would come another 5 minute round of chess, followed by another 800m race, the running advantage would accumulate and a player could win by gaining a lead greater than 1 minute or checkmating/winning on time against there opponent.