IME, the pedestrian has a Don't Walk sign if the car has a green left turn arrow.

If the car has a plain green light and is turning left, then they need to watch for pedestrians and for oncoming traffic.

There’s an intersection like this down the street from my house. The left arrow is barely long enough for three cars to go and it’s often backed up. Drivers are habituated to take the turn as fast as possible because the green arrow is so short. I’ve seen people almost get hit pretty often. I’ve seen drivers who yield get raged on because the green arrow expires before any cars get a chance to go.

The problem is that people tend to interpret a green light to mean "you can go" rather than "this specific light isn't telling you to stop". Hence all those signs that say something to the effect "turning cars yield to oncoming traffic on green" as a reminder. And the effect is likely worse for a green arrow, which implies having the right of way.