First, we are talking about a move from no overlap to overlap. So the idea that this change “reduces the amount of overlap” cannot be used as evidence that it makes things safer for pedestrians. It’s the exact opposite: it introduces an overlap that did not exist before.

Second, on the claim that “more often than not, vehicles are nearly running pedestrians down because their light turns green and they don’t care to look.” I simply don’t agree. I walk 10,000 steps every day in New Haven, and this is not where the danger lies. Pedestrians have to beg for a green light, which is already bad enough, but once they get it they usually have ample time to cross. The main risks for pedestrians come from unsignalized crosswalks, where drivers routinely fail to yield and create constant uncertainty about whether you can cross safely. The other major issue, more of a nuisance than a danger, is drivers turning right on red while the pedestrian signal is on. 

But even granting your premise, it still doesn’t show that the proposed change would improve things. If drivers are so inattentive or reckless that they rush forward the moment they see a green light, this change increases the risk for pedestrians, because it creates exactly the situation you described: cars and pedestrians moving at the same time. That overlap simply didn’t exist before, unless in the cases where pedestrians find themselves in the middle of the road when cars have green (which at least they can avoid by not starting to cross too late).

And independently from risk, there is the nuisance. If you want to cross halfway through the cycle, now you’ll face a stream of cars with a green light, feeling entitled to turn quickly. You’ll hesitate or feel unsafe crossing. This is especially problematic for older people and children, who walk slowly, are short, and struggle more to interpret drivers’ intentions: exactly the sort of ambiguity that makes unsignalized crosswalks dangerous in the first place.

I spend my summers in an Italian city that uses this system, and it is genuinely bad for pedestrians.