My guess would be that the house was in "disorder" and confusion/panic tonals with the injuries where the voices reflected that.
He heard you interacting with the family about what had occured when you got home and the signals were that alpha [ you ] was now home and involved.
As long as a family member was still inflecting panic, concern, etc and alpha was home now, Max would connect there was a "problem" and act accordingly in a submissive manner to alpha upon his physical presence [ usually waiting to see what alphas manner was and deferring to alpha and feeding off the "vibes" given one way or the other.
My guess, and it's only a guess as each dog is a little different [ they have personalities like us which vary from breed to breed and within their own breeds themselves ] that as long as there was commotion/panic/raised voices etc coming from family members he was going to defer to alpha and play the "wait and see".
Though you did not speak immediately to Max, he was painfully aware there was still a problem due to the tonals of the others. They get very submissive if they even "think" alpha is upset.
Most do not realize how well dogs can hear. As an example, my own Akita [ though no longer with us ] could be at the park and off lead playing a good distance [ say 50 yrds ] away and I'd call her name. She would keep her nose to the ground like she didn't hear me. As a test several times, I would call her name and she would not respond in the slightest manner to her name. I then would open a waxed paper bag like I was getting a doughnut hole out for her and she heard the bag rustling when opened and came running [ from 50 yrds out ]. The bag was hidden from her view, it was sound alone that got her attention.
She could hear the bag open but couldn't "hear" me call her name. Ya, right.
Once she had been to the park for any length of time, she knew when I called her we were going home. She didn't want to go so she ignored me like she didn't hear me.
I solved the problem by calling her once, if she did not respond and look up for further direction, I walked to her and put her on lead and in the car with a correction on the lead so she knew I was not happy about it. After several times of this at the park, she would look up and wait for the next signal/command. If I called her back, she started in my direction immediately.
She would play the game and try to make me call her twice [ to see if I really wanted her attention ]. That got old so I would call her once and if no response immediately, I'd start walking to her. As soon as I did that, she would come running [ unfortunately, too late as I continued to her and put the lead on with the correction so she knew I was unhappy ] and into the car. No words were spoken.
She would play the games until she knew she couldn't get away with them and would lose her freedom to roam. I would call her back and when she got there, let her go run again repeatedly everytime at the park [ daily ].
I did this so she knew that when I called her, she didn't necessarily get put into the car and go home.
She learned well over the years and I never had to touch her in anger. She knew with no words spoken I was displeased with her if I had to walk to her or move in her direction because she did not respond.
Dogs want/live to keep alpha happy once the rules have been established.
I think you are on the right track here. I would suggest this if I were in your shoes, but it's only a suggestion to consider.
Have the family go to a park with you [ or anywhere you can have some family time ]. Call him from off lead. Do what I did if there is no immediate response to you. Once he is "good to go" with you, have a family member call him. Same scenario, no response, go get him on lead. He'll learn pretty quick that he needs to pay attention to others in the house besides you.
I wish you all the luck with Max sir. I hope it all works out for you and the family. There is nothing like having a dog in your life if you like dogs at all. My Akita [ Akasha ] brought so much joy to my wife and I in so many different ways. She now rests on the mantle of the fireplace and will always be with us.
Respectfully
Brownie