It may also have to do with light traffic and not many people around in the wee hours of the morning. Most fighters seem to run on the streets. It would seem a good idea to be able to run without many people and cars around. That being said, I wonder why they just don't run on an outdoor track or around a park or athletic field? Why the streets?
When I saw Carlos Monzon, well after his retirement on March 24 1979 at the Sugar Ray Leonard vs Daniel Gonzales bout, Monzon looked trim, especially up close as we nodded up at each other in a sign of respect.
Waking up late is not good for you, your body feels better if you sleep early and wake up early. It is what humans did for ages!