As for why the dog form factor, I'm betting it'll be because a dog form factor makes for a much cheaper swarm than a tank form factor. Less mass to move about so a smaller power pack, too
Still, I share your skepticism of these robot dogs. They seem like a solution looking for a problem.
Edit: Why would a robo dog necessarily fall over? They could take a wide stance and essentially become a tripod if they need stability.
That's just what they do. Several Youtubers have fired guns off the robot dogs and even firing straight ahead can knock them over. Non autonomous tanks have navigated stairs just fine for the FBI and SWAT teams they do it all the time.
Maybe four legs provides better all-terrain capabilities at lower parts cost than a treaded vehicle?
The special forces ideal is small packages with large firepower. I can not imagine anything that better meets that definition than quadracopter scouts and robot dogs with missile launchers.
Falling over after the shot is irrelevant. Basic marksmanship training teaches recruits to pull the trigger until the shot going off surprises them. We only care about what happens next if we want a quick follow up shot, which is not possible with single shot rocket launchers.