Maybe if McCoy hadn't gone back in time, the homeless man would have slept off his drunkenness that night in the back room McCoy ultimately took refuge in. That would be a nice little bit of loose-end tying...
I would say this--you can't assume he would have died anyway, but clearly, history was unchanged by his death. In the absence of anything in the story to the contrary, we have to assume that. That leaves one of two options--one that man had no impact on the world, which is pretty hard no matter who you are, and they got very lucky. The other option is that McCoy clearly affected two lives, not one, when he traveled back in time. He caused the homeless man to die and he caused Edith to live. Spock and Kirk only focused on one event, not multiple, because they didn't know and didn't think about it. They kind of had one shot, and thanks to the tricorder, it became clear that Edith had to die. That was such a big event that Spock hadn't even considered there could have been more than one person impacted by McCoy.
Given that everyone who interacts with people will affect the world in a big way, the other option would have to be that by causing that homeless guy's death, McCoy actually helped set forth the events that led to the 23rd century as he knew it. Had he NOT traveled in time at all, that man could have impacted history in another way that would have had a massive effect on things.
You can even take it one step further. Spock was working with "stone knives and bear skins."
He didn't have the time nor the ability to fully analyze his tricorder.
How do we know that when Edith lived, Spock wasn't looking at a world that never had McCoy travel to it?
Think about it--there were two headlines--one that led to each timeline.
But nothing said which one was affected by McCoy.
Nothing said McCoy saved Edith's life.
How do we know that the timeline where the Nazis lost WWII, the real timeline, wasn't the one that unfolded through this episode?
As in, if McCoy DIDN'T travel in time, Edith never would have met Kirk, and never would have crossed the street trying to figure out how Kirk and McCoy knew each other?
There is no way to know for certain, but what if McCoy's time travel, which forced Kirk's and Spock's time travel, was the element that got Edith killed, which changed history, and caused the Nazis to lose a war that they actually would have won had Kirk, Spock and McCoy not caused a traffic accident that killed Edith?
Spock only showed us two scenarios--he never did say which was the one that involved interference from time travelers. He only ASSUMED the one where the Allies won WWII was the right one because that was HIS timeline.