@sidereal A few years ago someone pointed me at a fairly old paper that showed that the optimal behaviour for these was to change as soon as a pedestrian pushed the button. It had a lot of queueing theory modelling the different behaviours but roughly (and from memory):

If the road is busy, waiting doesn’t make it less busy, you’ll delay cars by the same amount, just a different set.

If the road is bursty, you’ll cause a little bubble of cars but they’ll typically be the same as at the next intersection. I’m a few situations, delaying cars a bit before an intersection can improve throughout. The only thing you need to do is make sure that a crossing light near an intersection doesn’t go red just as the intersection light goes green.

If the road is mostly empty, pedestrians will cross early and then the light will change later, where it has a higher probability of stopping a car.

It’s worth noting that, in the UK, zebra crossings have the property you want. It is illegal to enter a zebra crossing in a vehicle while people are on the crossing or waiting to cross. Legally, you can step out onto them without looking and it’s drivers’ responsibility to avoid you, though practically it’s worth making sure that oncoming cars have time to slow down and have seen you.