...or just light the "WALK" sign when the parallel street gets its green.
When the WALK sign is on, there's a minimum time required for pedestrian crossing that has to be allowed. Signals can switch faster and respond to traffic needs more quickly if they don't have to always block waiting for that crossing time.
IMO, the ideal solution is you add all-way crossing into every signal cycle. E.g. North/South traffic, then East/West traffic, then all-WALK (unless nobody pressed the WALK button). It's much safer - no worries about right-on-red or un-protected left turns hitting pedestrians, and pedestrians get the more efficient option to cross diagonally.
Certainly this would be a poor tradeoff in rural/suburban areas where most traffic cycles have no pedestrians or cyclists. Those use cases should use beg buttons.
But in a city where most traffic cycles include at least one pedestrian or cyclist then it seems like a useful tradeoff to increase convenience and accessibility for pedestrians and cyclists. What's more it may increase automobile efficiency as well: adding a couple Leading Pedestrian Intervals to every traffic cycle takes similar overall time than adding a full Barnes Dance[2] to half the traffic cycles. So if more than half of the traffic cycles include pedestrians crossing then everyone is probably better off with automatic walk signs on a leading pedestrian interval.
Of course if the city wants to really encourage walking they can do both and have the beg buttons trigger a Barnes dance for those who are young, old, nervous, or just need extra time to cross.
[1] https://highways.dot.gov/safety/proven-safety-countermeasure...
[2] what you describe above where all walk signs are on and all automobiles are stopped: https://en.wikipedia.org/wiki/Pedestrian_scramble