According to Elliotte Friedman of Sportsnet, Babcock's final straw wasn't exactly how he had painted asking players for photos of their families, but more more beyond that; take a look.
Babcock, in fact, had a meeting that took place away from team facilities where he took a players phone for ‘several minutes' and went through it, which, rightly caused a lot of concern.
In any case, taking an players phone without their permission is wrong, holding onto it for more than servers minutes makes it worse and lying about how the events took place makes it impossible for an NHL organization to keep you.
At the end of the day, Mike Babcock has no one else to blame for having to step down, but himself and he's more than likely ruined any hope he had of returning to an NHL bench.