Why is walking the salient technology that will enable this? Why not wheels, or treads, or flying?

Psychologically because it's life-like (uncanny valley).

Practically, because legs are superior to wheels when it comes to pursue a human target. It may be superior to flying as well indoor, to open doors, etc. And then there is the energy expenditure and additional mass constraint when airborne.

Research on legged robots is not just for 'fun', it's because they are better at going through obstacles and rough terrain than wheeled vehicles.

Part of it is because we already have flying/wheeling drones. Walking is just the last step of all terrain capabilities.

Flying can get you almost anywhere, but not quite. It is both very loud and has issues in enclosed spaces.

Wheels/tracks are very efficient at moving mass, but have issues crossing some materials such as exploded building internals like you would find in a war zone.

Walking/climbing is the last bit to get you inside enclosed 'human' spaces such as buildings and tunnels.

The war occurring now in Gaza would be an example area of what future battle would look like.... A target building it hit by a bomb and mostly destroyed. Smaller automated drones quickly fly over the area and assess visible targets and threats for follow up artillery. Multiple mobilized traced/wheeled heavily armored drones would address any residual ground level threats. Upon reaching the target building the larger drone vehicles would release multiple smaller 'walking/crawling' style drones to ascend the rubble, either addressing any remaining targets themselves, or reporting back to a command station for some other group to make a decision on what to do next. The final goal of these walking units is to probe deep into the rubble to ascertain if any tunnels survived the initial bombing, and if so to begin mapping any threats in said tunnels.

It's not necessarily walking, just the improvement in mobility which is one precursor to self-sufficient policing

Or obstacle avoidance and hints of advancing capabilities for learning and adapting? (Apparent) autonomy in general?

Sometimes when I watch videos like this I really wonder what the future is supposed to look like , heavily regulated I guess? All out chaos? Imagine what a criminal organization could do with a bunch of robots that are this agile? It's hard to imagine.

Personally,I think we'll get really good at killing robots and they'll get better at killing people (military).