That is extremely common at city intersections. The pedestrian has priority but the car also has a green light for a turn. I don't know about the exact timing of the sequence but there are many intersections where the pedestrian has a right to cross and a car has a right to turn (but must yield).

The same city (like many places) also has pedestrian crosswalks that aren't light controlled at all.

The cars need to watch for pedestrians. Lights don't necessarily means that cars obeying the lights never ever have to watch for pedestrians also crossing with the lights.

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.

Yeah, left turn on green always scares me as I have twice come close to hitting pedestrians that I just didn't see (at first) because my line of sight to them was blocked by the driver side pillar. Notably, both of these streets were three or more lanes in each direction, not sure if that adds a risk factor.

Huh? I've never seen this. If the drivers get a green turn arrow the pedestrians get a don't-walk. However, I see (and have been one) pedestrians who understand the light going anyway across the part of the road that's facing a red, treating it as two separate crossings. While this is not a problem I have also seen pedestrians going the other way start to cross when the oncoming pedestrians start--and they may be crossing against the green arrow. Same as you sometimes see drivers go because the lane next to them went but it was actually on a different traffic control.

As for the pedestrian hitting the button--I have never seen the button alter the sequence of the lights. The button does two things: Cause the light to cycle (in case it's sensor controlled and won't cycle if nobody comes along) and ensuring the cycle time is long enough for a pedestrian to cross. I've also seen hitting the button causing the button to indicate the status of the light (for the blind.)

There's a lot of ways to signal this.

With a left turn arrow, that should never be green when the turn conflicts with a pedestrian crossing that's on Walk. An arrow is an indicationn of a protected turn, where there should not be any conflicts.

But many intersections allow for an unprotected left turn, if safe, on a green ball in the forward direction. Sometimes there's a green and yellow arrow next to a green and yellow ball, with red on top in the middle of the two: you can usually make an unprotected left, if safe, if the green ball is showing, and if you have a green arrow, it's a protected left. I've also seen similar with a flashing yellow arrow indicating an unprotected left is permitted, if safe.

Of course, it's possible that the intersection was miswired or misconfigured. If the direction with the green arrow also had a green ball for forward, it may have been that the right side of that direction was supposed to have a Walk, but the signals were tragically misconfigured. I have experienced a case where wires were mixed up, but in the case I'm aware of, a bike button was configured as a left turn occupancy detector for another approach; inconvenient, but not dangerous.

In my city, the signal controller registers when a pedestrian pushes a button for the walk signal on a crossing that conflicts with a protected left turn. If it is fully protected, with a green arrow followed by red ball, then pedestrians are not given walk signals during that turning phase but prior during the through phase in their direction. If it is protected-permissive, when motorists see a green arrow followed by a green ball, then the pedestrian gets their walk signal ahead of the turning traffic, so they can clear before the left turning vehicles intersect their crosswalk. The collision you saw is the same issue that happens with right turn on red. The pedestrian has a walk, the car behind them turns into the pedestrian, even though they have a better view than a left turning vehicle of the nearby person. One alternative is to give separate phases to each traffic movement, such as the pedestrian scramble where all directions of pedestrians go with all vehicles stopped. The issue with delaying the left turn while the pedestrians go first is it increases cycle time or gives a greater percentage of cycle time to lower volume side road traffic, which tends to reduce the service level for most motorists waiting at conflicting red signals for these movements to complete. Lower service levels lead to more complaints, especially when there are thousands of vehicles per day vs. 100 pedestrian crossings, but Traffic Engineers have to fight for those pedestrians using what is often a difficult compromise.