There are two pedestrian crosswalks out of dozens in the town I live in that work this way. Confuses the crap out of the pedestrians and the drivers, as they aren't expecting it to work that way.

All the other intersections are timed so drivers and pedestrians go at the same time. If we switched all the intersections to allow pedestrians to go first, I think that would be safer. Getting city council to do anything is another issue.

In the UK pedestrians will never have a green light while cars have a green light and the timings wait until pedestrians will have finished crossing before allowing cars. There might be some slow walkers or people who crossed without green (it's dangerous but not illegal) but everything is done to ensure neither pedestrians nor cars are told to go while the other isn't told to stop.

I have a question about these. Where I live there's typically a green light for pedestrians, red for cars at the start of the cycle. Then it switches to green for cars, flashing for pedestrians, which means that cars have to give way, pedestrians may finish crossing, but no pedestrians may start crossing.

I am a fast walker, in a hurry, confident at dodging cars, and this is a slow intersection. I arrive at the crosswalk when the pedestrian light is flashing but cross anyway, I know I have plenty of time. Turning car honks on having to give way to me.

Which rule takes priority? I think that even though I'm technically breaking the rules, the car should still have to follow the rules and give way to pedestrians.

Pedestrians very often get the green walk sign at the same time as cars in the same direction get the green straight ahead sign that implicitly allows right turns as well. In that scenario, the cars are often located just behind the pedestrian's peripheral vision, and the cars are looking up and ahead at the light (or maybe to their left if they were previously hoping to execute a right on red and waiting for a gap in traffic), so it becomes basically exactly what you were talking about with bikes. Cars turning across a "lane" of pedestrian traffic with neither party having good visibility of the other. There are a couple of solutions/mitigations in use for this problem, dedicated "all-red" pedestrian cycles, protected intersections that move vulnerable road users further forward to be more visible, and advanced pedestrian greens that make cars wait until pedestrians are already in the intersection and more visible before getting the go ahead. Or, if your city is like mine and car-centric, they might stick up a yellow sign on the opposite light post across the stroad that says "pedestrians yield to turning vehicles" in text that is just barely legible from across the street at an intersection that has audible wait and walk indicators for blind people who definitely can't read that sign and will thus be endangered for not getting the memo (not that car drivers are reading it either, so several considerate drivers will wave the pedestrians forward, further confusing the right of way situation). Fun!

In short, every turn you make as a driver should be accompanied by a check for vulnerable road users like cyclists and pedestrians because our infrastructure will not necessarily put them in a place that is easily visible to you

Pedestrians have the right of way and get the walk signal on the green, is it really that foolish? Also children, elderly, visually/auditory impaired people all also cross the street. It is 100% the responsibility of the driver to ensure the intersection is clear before traveling through it, this includes clear of pedestrians.